QB 

201 


;-NRLF 


PRACTICAL  ASTRONOMY 
FOR  ENGINEERS 


PRACTICAL  ASTRONOMY 
FOR   ENGINEERS 


BY 


FREDERICK  HANLEY  SEARES 

Professor  of  Astronomy  in  the  University  of  Missouri 
and  Director  of  the  Laws  Observatory 


COLUMBIA,    MISSOURI 

THE  E.  W.  STEPHENS  PUBLISHING  COMPANY 
1909 


COPYRIGHT,  1909 

BY 
FREDERICK    HANLEY   SEARES 


PREFACE 

The  following  pages  represent  the  result  of  several  years'  experience  in 
presenting  to  students  of  engineering  the  elements  of  Practical  Astronomy. 
Although  the  method  and  the  extent  of  the  discussion  have  been  designed  to 
meet  the  specialized  requirements  of  such  students,  it  is  intended  that  the 
work  shall  also  serve  as  an  introduction  for  those  who  desire  a  broader  knowl- 
edge of  the  subject. 

The  order  of  treatment  and  the  methods  proposed  for  the  solution  of  the 
various  problems  have  been  tested  sufficiently  to  establish  their  usefulness; 
and  yet  the  results  are  to  be  regarded  as  tentative,  for  they  possess  neither  the 
completeness  nor  the  consistency  which,  it  is  hoped,  will  characterize  a  later 
edition.  The  volume  is  incomplete  in  that  it  includes  no  discussion  of  the 
principles  and  methods  of  the  art  of  numerical  calculation — a  question  funda- 
mental for  an  appreciation  of  the  spirit  of  the  treatment.  Difficulties  inherent 
in  this  defect  may  be  avoided  by  a  careful  examination  of  an  article  on 
numerical  calculation  which  appeared  in  Popular  Astronomy,  1908,  pp.  349-367, 
and  in  the  Engineering  Quarterly  of  the  University  of  Missouri,  v.  2,  pp.  171-192. 
The  final  edition  will  contain  this  paper,  in  a  revised  form,  as  a  preliminary 
chapter.  The  inconsistencies  of  the  work  are  due  largely  to  the  fact  that  the 
earlier  pages  were  in  print  before  the  later  ones  were  written,  and  to  the 
further  fact  that  the  manuscript  was  prepared  with  a  haste  that  permitted  no 
careful  interadjustment  and  balancing  of  the  parts. 

The  main  purpose  of  the  volume  is  an  exposition  of  the  principal  methods 
of  determining  latitude,  azimuth,  and  time.  Generally  speaking,  the  limit  of 
precision  is  that  corresponding  to  the  engineer's  transit  or  the  sextant.  Though 
the  discussion  has  thus  been  somewhat  narrowly  restricted,  an  attempt  has  been 
made  to  place  before  the  student  the  means  of  acquiring  correct  and  complete 
notions  of  the  fundamental  conceptions  of  the  subject.  But  these  can  scarcely 
be  attained  without  some  knowledge  of  the  salient  facts  of  Descriptive 
Astronomy.  For  those  who  possess  this  information,  the  first  chapter  will 
serve  as  a  review;  for  others,  it  will  afford  an  orientation  sufficient  for  the 
purpose  in  question.  Chapter  II  blocks  out  in  broad  lines  the  solutions  of  the 
problems  of  latitude,  azimuth,  and  time.  The  observational  details  of  these 
solutions,  with  a  few  exceptions,  are  presented  in  Chapter  IV,  while  Chapters 
V-VII  consider  in  succession  the  special  adaptations  of  the  fundamental 
formulae  employed  for  the  reductions.  In  each  instance  the  method  used  in 
deriving  the  final  equations  originates  in  the  principles  underlying  the  subject 
of  numerical  calculation.  Chapter  III  is  devoted  to  a  theoretical  considera- 
tion of  the  subject  of  time. 

It  is  not  customary  to  introduce  historicat  data  into  texts  designed  for 
the  use  of  professional  students;  but  the  author  has  found  so  much  that  is 


198988 


vi  PREFACE 

helpful  and  stimulating  in  a  consideration  of  the  development  of  astronomical 
instruments,  methods,  and  theories  that  he  is  disposed  to  offer  an  apology  for 
the  brevity  of  the  historical  sections  rather  than  to  attempt  a  justification  of 
their  introduction  into  a  work  mainly  technical  in  character.  To  exclude 
historical  material  from  scientific  instruction  is  to  disregard  the  most  effective 
means  of  giving  the  student  a  full  appreciation  of  the  significance  and  bearing 
of  scientific  results.  Brief  though  they  are,  it  is  hoped  that  these  sections 
may  incline  the  reader  toward  wider  excursions  into  this  most  fascinating  field. 

-The  numerical  solutions  for  most  of  the  examples  have  been  printed  in 
detail  in  order  better  to  illustrate  both  the  application  of  the  formulae  involved 
and  the  operations  to  be  performed  by  the  computer.  Care  has  been  taken  to 
secure  accuracy  in  the  text  as  well  as  in  the  examples,  but  a  considerable 
number  of  errors  have  already  been  noted.  For  these  the  reader  is  referred 
to  the  list  of  errata  on  page  132. 

The  use  of  the  text  should  be  supplemented  by  a  study  of  the  prominent 
constellations.  For  this  purpose  the  "Constellation  Charts"  published  by  the 
editor  of  Popular  Astronomy,  Northfield,  Minnesota,  are  as  serviceable  as  any, 
and  far  less  expensive  than  the  average. 

My  acknowledgments  are  due  to  Mr.  E.  S.  Haynes  and  Mr.  Harlow 
Shapley,  of  the  Department  of  Astronomy  of  the  University  of  Missouri,  for 
much  valuable  assistance  in  preparing  the  manuscript,  in  checking  the  calcu- 
lations, and  in  reading  the  proofs. 

F.  H.  SEARES. 
LAWS  OBSERVATORY, 
UNIVERSITY  OF  MISSOURI, 
June,  1909. 


CONTENTS 


CHAPTER  I 

INTRODUCTION— CELESTIAL    SPHERE—COORDINATES 

PAGE. 

1 .  The  results  of  astronomical  investigations   '.  1 

2.  The  apparent  phenomena  of  the  heavens  4 

3.  Relation  of  the  apparent  phenomena  to  their  interpretation  5 

4.  Relation  of  the  problems  of  practical  astronomy  to  the  phenomena  of  the  heavens  7 

5.  Coordinates  and  coordinate  systems   , 8 

6.  Characteristics  of  the  three  systems.     Changes  in  the  coordinates  10 

7.  Summary.     Method  of  treating  the  corrections  in  practice  ._ 15 

8.  Refraction 16 

9.  Parallax    18 

CHAPTER  II 

FORMULAE    OF    SPHERICAL    TRIGONOMETRY— TRANSFORMATION    OF 
COORDINATES— GENERAL  DISCUSSION  OF  PROBLEMS 

1 0.  The  fundamental  formulae  of  spherical  trigonometry   21 

11.  Relative  positions  of  the  reference  circles  of  the  three  coordinate  systems  ....  23 

12.  Transformation  of  azimuth  and  zenith  distance  into  hour  angle  and  declina- 
tion      25 

13.  Transformation  of  hour  angle  and   declination   into  azimuth  and  zenith  dis- 
tance   29 

14.  Transformation  of  hour  angle  into  right  ascension,  and  vice  versa  29 

1 5.  Transformation  of  azimuth  and  altitude  into  right  ascension  and  declination, 
and  vice  versa   31 

16.  Given  the  latitude  of  the  place,  and  the  declination  and  zenith  distance  of  an  , 
object,  to  find  its  hour  angle,  azimuth,  and  parallactic  angle  31 

17.  Application  of  transformation  formulae  to  the  determination  of  latitude,  azimuth, 
and    time    32 

CHAPTER  III 

TIME  AND  TIME   TRANSFORMATION 

1 8.  The  basis  of  time  measurement   36 

1 9.  Apparent,  or  true,   solar  time    36 

20.  Mean  solar  time  > 36 

21 .  Sidereal  time    ' 37 

22.  The  tropical  year  38 

23.  The  calendar. 38 

24.  Given  the  local  time  at  any  point,  to  find  the  corresponding  local  time  at  any 
other   point    39 

25.  Given  the  apparent  solar  time  at  any  place,  to  find  the  corresponding  mean 
solar   time,   and   vice   versa    40 

26.  Relation  between  the  values  of  a  time  interval  expressed  in  mean  solar  and 
sidereal  units   42 

27.  Relation  between  mean  solar  time  and  the  corresponding  sidereal  time  44 

28.  The  right  ascension  of  the  mean  sun  and  its  determination  44 

29.  Given  the  mean  solar  time  at  any  instant  to  find  the  corresponding  sidereal 
time 47 

30.  Given  the  sidereal  time  at  any  instant  to  find  the  corresponding  mean  solar 
time    ,  48 


viii  CONTENTS 

CHAPTER  IV 

INSTRUMENTS    AND    THEIR   USE 

PAGE. 

3 1 .  Instruments  used  by  the  engineer 50 

TIMEPIECES 

32.  Historical     50 

33.  Error  and  rate 51 

34.  -Comparison   of  timepieces    52 

35.  The  care  of  timepieces   58 

THE  ARTIFICIAL  HORIZON 

36.  Description  and  use   59 

THE  VERNIER 

37.  'Description  and  theory 59 

38.  Uncertainty  of  the  result 60 

THE  ENGINEER'S  TRANSIT 

39.  Historical     61 

40.  Influence  of  imperfections  of  construction  and  adjustment   62 

4 1 .  Summary  of  the  preceding  section   71 

42.  The  level    71 

43.  Precepts  for  the  use  of  the  striding  level  72 

44.  Determination  of  the  value  of  one  division  of  a  level  73 

45.  The  measurement  of  vertical  angles   77 

46.  The  measurement  of  horizontal  angles  80 

47.  The   method   of   repetitions 81 

THE  SEXTANT 

48.  Historical  and  descriptive    85 

49.  The  principle  of  the  sextant  86 

50.  Conditions  fulfilled  by  the  instrument   87 

5 1 .  Adjustments  of  the   sextant    88 

52.  Determination  of  the  index  correction   89 

53.  Determination  of  eccentricity  corrections   90 

54.  Precepts  for  the  use  of  the  sextant  , 91 

55.  The  measurement  of  altitudes 91 

CHAPTER  V 

THE  DETERMINATION  OF  LATITUDE 

56.  Methods     95 

1.    MERIDIAN  ZENITH  DISTANCE 

57.  Theory 96 

58.  Procedure 96 

2.     DIFFERENCE  OF  MERIDIAN  ZENITH  DISTANCES — TALCOTT'S  METHOD 

59.  Theory 97 

60.  Procedure   98 

3.      ClRCUMMERIDIAN    ALTITUDES 

61.  Theory 99 

62.  Procedure   .                                                                                                                          .  101 


CONTENTS  ix 

4.    ZENITH  DISTANCE  AT  ANY  HOUB  ANGLE 
63.    Theory     


PAGE. 
.  .    102 
.   103 


64.  Procedure    

5.    ALTITUDE  OF  POLABIS 

104 

65.  Theory     

66.  Procedure  


67.  Influence  of  an  error  in  time  ..............................................  J 

CHAPTER  VI 

THE  DETERMINATION  OF  AZIMUTH 

68.  Methods    ................................................... 

1.    AZIMUTH  OF  THE  SUN 

1  AQ 

69.  Theory     ......................................................... 

70.  Procedure   ..................................................................  * 

2.    AZIMUTH  OF  A  CIRCUMPOLAB  STAB  AT  ANY  HOUB  ANGLE 

71.  Theory     ....................................................................  ™ 

72.  Procedure   .................................................................. 

3.     AZIMUTH  FROM  AN  OBSERVED  ZENITH  DISTANCE 

113 


73.  Theory 

74.  Procedure 


75.  Azimuth  of  a  mark  1 

76.  Influence  of  an  error  in  the  time  H* 

CHAPTER  VII 

THE  DETERMINATION  OF  TIME 

77.  Methods    116 

1.    THE  ZENITH  DISTANCE  METHOD 

78.  Theory     J 

79.  Procedure   118 

2.     THE  METHOD  OF  EQUAL  ALTITUDES 

80.  Theory     1 

8 1 .  Procedure   ^ 

3.    THE  MERIDIAN  METHOD 

82.  Theory  1 

83 .  Procedure    123 

4.    THE  POLABIS  VERTICAL  CIRCLE  METHOD 
SIMULTANEOUS  DETERMINATION  OF  TIME  AND  AZIMUTH 

84.  Theory     126 

85.  Procedure 


129 


ERRATA  132 


INDEX 


133 


PRACTICAL  ASTRONOMY 
FOR   ENGINEERS 

CHAPTER  I 

INTRODUCTION— CELESTIAL    SPHERE— COORDINATES. 

1.  The  results  of  astronomical  investigations. — The  investigations  of 
the  astronomer  have  shown  that  the  universe  consists  of  the  sun,  its  attendant 
planets,  satellites,  and  planetoids;  of  comets,  meteors,  the  stars,  and  the 
nebulae.  The  sun,  planets,  satellites,  and  planetoids  form  the  solar  system, 
and  with  these  we  must  perhaps  include  comets  and  meteors.  The  stars 
and  nebulae,  considered  collectively,  constitute  the  stellar  system. 

The  sun  is  the  central  and  dominating  body  of  the  solar  system.  It  is 
an  intensely  heated  luminous  mass,  largely  if  not  wholly  gaseous  in  consti- 
tution. The  planets  and  planetoids,  which  are  relatively  cool,  revolve  about 
the  sun.  The  satellites  revolve  about  the  planets.  The  paths  traced  out  in 
the  motion  of  revolution  are  ellipses,  nearly  circular  in  form,  which  vary 
slowly  in  size,  form,  and  position.  One  focus  of  each  elliptical  orbit  coin- 
cides with  the  center  of  the  body  about  which  the  revolution  takes  place. 
Thus,  in  the  case  of  the  planets  and  planetoids,  one  of  the  foci  of  each  orbit 
coincides  with  the  sun,  while  for  the  satellites,  the  coincidence  is  with  the 
planet  to  which  they  belong.  In  all  cases  the  form  of  the  path  is  such  as 
would  be  produced  by  attractive  forces  exerted  mutually  by  all  members  of 
the  solar  system  and  varying  in  accordance  with  the  Newtonian  law  of 
gravitation.  In  addition  to  the  motion  of  revolution,  the  sun,  planets,  and 
some  of  the  satellites  at  least,  rotate  on  their  axes  with  respect  to  the  stars. 

The  planets  are  eight  in  number.  In  order  from  the  sun  they  are : 
Mercury,  Venus,  Earth,  Mars,  Jupiter,  Saturn,  Uranus,  and  Neptune.  Their 
distances  from  the  sun  range  from  thirty-six  million  to  nearly  three  thousand 
million  miles.  Their  diameters  vary  from  about  three  thousand  to  nearly 
ninety  thousand  miles.  Nevertheless,  comparatively  speaking,  they  are  small, 
for  their  collective  mass  is  but  little  more  than  one  one-thousandth  that  of 
the  sun. 

The  planetoids,  also  known  as  small  planets  or  asteroids,  number  six 
hundred  or  more,  and  relatively  to  the  planets,  are  extremely  small  bodies — 
so  small  that  they  are  all  telescopic  objects  and  many  of  them  can  be  seen 
only  with  large  and  powerful  instruments.  Most  of  them  are  of  compara- 
tively recent  discovery,  and  a  considerable  addition  to  the  number  already 
known  is  made  each  year  as  the  result  of  new  discoveries.  With  but  few 
exceptions  their  paths  lie  between  the  orbits  of  Mars  and  Jupiter. 

The  only  satellite  requiring  our  attention  is  the  moon.  This  revolves 
about  the  earth  with  a  period  of  about  one  month,  and  rotates  on  its  axis 
once  during  each  revolution.  Although  one  of  the  smaller  bodies  of  the 
solar  system  it  is,  on  account  of  its  nearness,  one  of  the  most  striking. 

l 


PRACTICAL  ASTRONOMY 


The  solar  and  stellar  systems  are  by  no  means  coordinate  parts  of  the 
universe.  On  the  contrary,  the  former,  vast  as  it  is,  is  but  an  insignificant 
portion  of  the  latter,  for  the  sun  is  but  a  star,  not  very  different  on  the 
average  from  the  other  stars  whose  total  number  is  to  be  counted  by  hun- 
dreds of  millions;  and  the  space  containing  the  entire  solar  system,  includ- 
ing sun,  planets,  satellites,  and  planetoids,  is  incredibly  small  as  compared 
with  that  occupied  by  the  stellar  system.  To  obtain  a  more  definite  notion 
of  the  relative  size  of  the  two  .systems  consider  the  following-  illustration : 
Let  the  various  bodies  be  represented  by  small  spheres  whose  diameters 
and  mutual  distances  exhibit  the  relative  dimensions  and  distribution  through 
space  of  the  sun,  planets,  and  stars.  We  shall  thus  have  a  rough  model  of 
the  universe,  and  to  make  its  dimensions  more  readily  comprehensible  let 
the  scale  be  fixed  by  assuming  that  the  sphere  representing  the  sun  is  two 
feet  in  diameter.  The  corresponding  diameters  of  the  remaining  spheres  and 
their  distances  from  the  central  body  are  shown  by  the  following  table. 


OBJECT 

DIAMETER 

DISTANCE 

Sun 

2  feet 

Mercury 

0.08  inch 

83  feet 

Venus 

0.21  inch 

155  feet 

Earth 

0.22  inch 

215  feet 

Mars 

o.  12  inch 

327  feet 

Jupiter 

2.42  inch 

1116  feet 

Saturn 

2.  02  inch 

2048  feet 

Uranus 

0.97  inch 

4118  feet 

Neptune 

0.91  inch 

6450  feet 

Nearest  Star 

Unknown 

nooo  miles 

Tt  will  be  seen  that  the  distance  of  the  outermost  planet  from  the  sun 
is  represented  in  the  model  by  about  a  mile  and  a  quarter.  On  the  same 
scale,  the  distance  of  the  nearest  star,  the  only  one  included  in  the  table, 
is  approximately  equal  to  one-half  the  circumference  of  the  earth.  When 
it  is  remembered  that  this  object  is  but  one  of  perhaps  two  hundred  million 
stars,  the  vast  majority  of  which  are  probably  at  least  one  hundred  times 
more  distant,  and  further  that  each  of  these  stars  is  a  sun  as  our  owin  sun, 
the  very  subordinate  position  of  the  solar  system  becomes  strikingly  ap- 
parent. 

The  fact  that  the  sun  is  similar  in  size  and  chemical  composition  to 
millions  of  other  stars  at  once  raises  the  question  as  to  whether  they  too 
are  not  provided  with  attendant  systems  of  planets  and  satellites.  A  de- 
finite answer  is  wanting,  although  analogy  suggests  that  such  may  well  be 
the  case.  Bodies  no  larger  than  the  planets  and  shining  only  by  reflected 
light  would  be  quite  invisible,  even  in  the  most  powerful  telescopes,  when 
situated  at  distances  comparable  with  those  separating  us  from  the  stars. 


,  INTRODUCTION  3 

We  do  know,  however,  that  in  many  instances  two  or  more  stars  situated 
relatively  near  each  other  revolve  about  their  common  center  of  gravity  thus 
forming  binary  or  multiple  systems.  The  discovery  and  study  of  these 
systems  constitutes  one  of  the  most  interesting-  and  important  lines  of  modern 
astronomical  investigation. 

The  distances  separating  the  various  members  of  the  solar  system  are 
such  that  the  motions  of  the  planets  and  planetoids  with  respect  to  the  sun, 
and  of  the  satellites  relative  to  their  primaries,  produce  rapid  changes  in 
their  positions  as  seen  from  the  earth.  The  stars  are  also  in  motion  and 
the  velocities  involved  are  very  large,  amounting  occasionally  to  a  hundred 
miles  or  more  per  second  of  time,  but  to  the  observer  on  the  earth,  their 
relative  positions  remain  sensibly  unchanged.  The  distances  of  these  ob- 
jects are  so  great  that  it  is  only  when  the  utmost  refinement  of  observation 
is  employed  and  the  measures  are  continued  for  months  and  years,  that  any 
shift  in  position  can  be  detected  even  for  those  which  move  most  rapidly. 
With  minor  exceptions,  the  configuration  of  the  constellations  is  the  same 
as  it  was  two  thousand  years  ago  when  the  observations  upon  which  are 
based  the  earliest  known  record  of  star  positions  were  made. 

To  the  casual  observer  there  is  not  a  great  deal  of  difference  in  the  ap- 
pearance of  the  stars  and  the  planets.  The  greater  size  and  luminosity  of 
the  former  is  offset  by  their  greater  distance.  In  ancient  times  the  funda- 
mental difference  between  them  was  not  known,  and  they  were  distinguished 
only  by  the  fact  that  the  planets  change  their  positions,  while  relatively  to 
each  other  the  stars  are  apparently  fixed.  In  fact  the  word  planet  means 
literally,  a  moving  or  wandering  star,  while  what  appeared  to  the  early  ob- 
servers as  the  distinguishing  characteristic  of  the  stars  is  shown  by  the  fre- 
quent use  of  the  expression  fixed  stars. 

The  nebulae  are  to  be  counted  by  the  hundreds  of  thousand's.  They  con- 
sist of  widely  extended  masses  of  luminous  gas,  apparently  of  simple  chemical 
composition.  They  are  irregularly  distributed  throughout  the  heavens,  and 
present  the  greatest  imaginable  diversity  of  form,  structure,  and  brightness. 
Minute  disc  like  objects,  rings,  double  branched  spirals,  and  voluminous 
masses  of  extraordinarily  complex  structure,  some  of  which  resemble  closely 
the  delicate  high-lying  clouds  of  our  own  atmosphere,  are  to  be  found  among 
them.  The  brightest  are  barely  visible  to  the  unaided  eye,  while  the  faintest 
tax  the  powers  of  the  largest  modern  telescopes.  Their  distances  are  of  the 
same  order  of  magnitude  as  those  of  the  stars,  and,  indeed,  there  appears 
to  be  an  intimate  relation  connecting  these  two  classes  of  objects,  for  there 
is  evidence  indicating  that  the  stars  have  been  formed  from1  the  nebulae 
through  some  evolutionary  process  the  details  of  which  are  as  yet  not  fully 
understood. 

The  preceding  paragraphs  give  the  barest  outline  of  the  interpretation 
which  astronomers  have  been  led  to  place  upon  the  phenomena  of  the 
heavens.  The  development  of  this  conception  of  the  structure  of  the  universe 
forms  the  major  part  of  the  history  of  astronomy  during  the  last  four  cen- 


4  PRACTICAL  ASTRONOMY 

turies.      Many  have   contributed   toward   the    elaboration   of   its   details,   but 
its  more  significant  features  are  due  to  Copernicus,  Kepler,  and  Newton. 

Although  the  scheme  outlined  above  is  the  only  theory  thus  far  formu- 
lated which  satisfactorily  accounts^for  the  celestial  phenomena  in  their  more 
intricate  relations,  there  is  another  conception  of  the  universe,  one  far  earlier 
in  its  historical  origin,  which  also  accounts  for  the  more  striking  phenomena. 
This  theory  bears  the  name  of  the  Alexandrian  astronomer  Ptolemy,  and, 
as  its  central  idea  is  immediately  suggested  b}'  the  most  casual  examination 
of  the  motions  of  the  celestial  bodies,  we  shall  now  turn  to  a  consideration 
of  these  motions  and  the  simple,  elementary  devices  which  can  be  used  for 
their  description. 

2.  The  apparent  phenomena  of  the  heavens. — The  observer  who  goes 
forth  under  the  star-lit  sky  finds  himself  enclosed  by  a  hemispherical  vault 
of  blue  which  meets  in  the  distant  horizon  the  seemingly  flat  earth  upon 
which  he  stands.  The  surface  of  the  vault  is  strewn  with  points  of  light 
of  different  brightness,  whose  number  depends  upon  the  transparency  of  the 
atmosphere  and  the  brightness  of  the  moon,  but  is  never  more  than  two  or 
three  thousand.  A  fewi  hours  observation  shows  that  the  positions  of  the 
points  are  slowly  shifting  in  a  peculiar  and  definite  manner.  Those  in  the 
east  are  rising  from  the  horizon  while  those  in  the  west  are  setting.  Those 
in  the  northern  heavens  describe  arcs  of  circles  in  a  counter-clockwise  di- 
rection about  a  common  central  point  some  distance  above  the  horizon. 
Their  distances  from  each  other  remain  unchanged.  The  system  moves  as 
a  whole. 

The  phenomenon  can  be  described  by  assuming  that  each  individual 
point  is  fixed  to  a  spherical  surface  which  rotates  uniformly  from'  east  to 
west  about  an  axis  passing  through  the  eye  of  the  observer  and  the  central 
point  mentioned  above.  The  surface  to  which  the  light-points  seem  at- 
tached is  called  the  Celestial  Sphere.  Its  radius  is  indefinitely  great.  Its 
period  of  rotation  is  one  day,  and  the  resulting  motion  of  the  celestial  bodies 
is  called  the  Diurnal  Motion  or  Diurnal  Rotation. 

The  daylight  appearance  of  the  heavens  i's  not  unlike  that  of  the  night 
except  that  the  sun,  moon,  and  occasionally  Venus,  are  the  only  bodies  to 
be  seen  in  the  celestial  vault.  They  too  seem  to  be  carried  along  with  the 
celestial  sphere  in  its  rotation,  rising  in  the  east,  descending  toward  the 
west,  and  disappearing  beneath  the  horizon  only  to  rise  again  in  the  east ; 
but  if  careful  observations  be  made  it  wall  be  seen  that  these  bodies  can 
not  be  thought  of  as  attached  to  the  surface  of  the  sphere,  a  fact  most  easily 
verified  in  the  case  of  the  moon.  Observations  upon  successive  nights  show 
that  the  position  of  this  object  changes  with  respect  to  the  stars.  A  con- 
tinuation of  the  observations  will  show  that  it  apparently  moves  eastward 
over  the  surface  of  the  sphere  along  a  great  circle  at  such  a  rate  that  an 
entire  circuit  is  completed  in  about  one  month.  A  similar  phenomenon  in 
the  case  of  the  sun  manifests  itself  by  the  fact  that  the  time  at  which  any 
o-iven  star  rises  does  not  remain  the  same,  but  occurs  some  four  minutes 


INTRODUCTION  5 

earlier  for  each  successive  night.  A  star  rising  two  hours  after  sunset  on  a 
given  night  will  rise  approximately  lh  56m  after  sunset  on  the  following 
night.  The  average  intervals  for  succeeding  nights  will  be  lh  52m,  lh  48m,  lh 
44m,  etc.,  respectively.  That  the  stars  rise  earlier  on  successive  nights  shows 
that  the  motion  of  the  sun  over  the  sphere  is  toward  the  east.  Its  path  is  a 
great  circle  called  the  Ecliptic.  Its  motion  in  one  day  is  approximately  one 
degree,  which  corresponds  to  the  daily  change  of  four  minutes  in  the  time 
of  rising  of  the  stars.  This  amount  varies  somewhat,  being  greatest  in 
January  and  least  in  July,  but  its  average  is  such  that  a  circuit  of  the 
sphere  is  completed  in  one  year.  This  motion  is  called  the  Annual  Motion 
of  the  Sun. 

With  careful  attention  it  will  be  found  that  a  few  of  the  star-like  points 
of  light,  half  a  dozen  more  or  less,  are  exceptions  to  the  general  rule  which 
rigidly  fixes  these  objects  to  the  surface  of  the  celestial  sphere.  These  are 
the  planets,  the  wandering  stars  of  the  ancients.  Their  motions  with  respect 
to  the  stars  are  complex.  They  have  a  general  progressive  motion  toward 
the  east,  but  their  paths  are  looped  so  that  there  are  frequent  changes  in  di- 
rection and  temporary  reversals  of  motion.  Two  of  them,  Mercury  and 
Venus,  never  depart  far  from  the  sun,  oscillating  from  one  side  to  the  other  in 
paths  which  deviate  but  little  from  the  ecliptic.  The  paths  of  the  others  also 
lie  near  the  ecliptic,  but  the  planets  themselves  are  not  confined  to  the 
neighborhood  of  the  sun. 

The  sun,  moon,  and  the  planets  therefore  appear  to  move  over  the  surface 
of  the  celestial  sphere  with  respect  to  the  stars,  in  paths  which  lie  in  or  near 
the  ecliptic.  The  direction  of  motion  is  opposite,  in  general,  to  that  of  the 
diurnal  rotation.  The  various  motions  proceed  quite  independently.  While 
the  sun,  moon,  and  planets  move  over  the  surface  of  the  sphere,  the  sphere 
itself  rotates  on  its  axis  with  a  uniform  angular  velocity. 

These  elementary  facts  are  the  basis  upon  which  the  theory  of  Ptolemy 
was  developed.  It  assumes  the  earth,  fixed  in  position,  to  be  the  central 
body  of  the  universe.  It  supposes  the  sun,  moon,  and  planets  to  revolve 
about  the  earth  in  paths  which  are  either  circular  or  the  result  of  a  com- 
bination of  uniform  circular  motions;;  and  regards  the  stars  as  attached  to 
the  surface  of  a  sphere,  which,  concentric  with  the  earth  and  enclosing  the 
remaining  members  of  the  system,  rotates  from  east  to  west,  completing  a 
revolution  in  one  day. 

3.  Relation  of  the  apparent  phenomena  to  their  interpretation. — The  re- 
lation of  the  apparent  phenomena  to  the  conception  of  Ptolemy  is  obvious, 
and  their  connection  with  the  scheme  outlined  in  Section  1  is  not  difficult  to 
trace.  The  celestial  sphere  rs  purely  an  optical  phenomenon  and  has  no  real 
existence.  The  celestial  bodies  though  differing  greatly  in  distance  are  all 
so  far  from  the  observer  that  the  eye  fails  to  distinguish  any  difference  in 
their  distances.  The  blue  background  upon  which  they  seem  projected  is 
due  partly  to  reflection,  and  party  to  selective  absorption  of  the  light  rays 
by  the  atmosphere  surrounding  the  earth.  As  already  explained,  the  stars 


6  PRACTICAL  ASTRONOMY 

are  so  distant  that,  barring  a  few  exceptional  cases,  their  individual  motions 
produce  no  sensible  variation  in  their  relative  positions,  and,  even  for  the 
exceptions,  the  changes  are  almost  vanishingly  small.  On  the  other  hand, 
the  sun,  planets,  and  satellites  are  relatively  near,  and  their  motions  produce 
marked  changes  in  their  mutual  distances  and  in  their  positions  with  respect 
to  the  stars.  The  annual  motion  of  the  sun  in  the  ecliptic  is  but  a  reflection 
of  the  motion  of  the  earth  in  its  elliptical  orbit  about  the  sun.  The  monthly 
motion  of  the  moon  is  a  consequence  of  its,  revolution  about  the  earth,  and 
the  complex  motions  of  the  planets  are  due,  partly  to  their  own  revolutions 
about  the  sun,  and  partly  to  the  rapidly  shifting  position  of  the  observer. 
Finally,  the  diurnal  rotation  of  the  celestial  sphere,  which  at  first  glance 
seems  to  carry  with  it  all  the  celestial  bodies,,  is  but  the  result  Of  the  axial 
rotation  of  the  earth. 

In  so  far  as  the  more  obvious  phenomena  of  the  heavens  are  concerned 
there  is  no  contradiction  involved  in  either  of  the  conceptions  which  have 
been  devised  for  the  description  of  their  relations.     That  such  is  the  case 
arises  from  the  fact  that  we  are  dealing  with  a  question  concerning  changes 
of  relative  distance  and  direction.     Given  two  points,  A  and  B,  we  can  de- 
scribe the  fact  that  their  distance  apart,  and  the  direction  of  the  line  joining 
them,  are  changing,  in  either  of  two  ways.     We  may  think  of  A  as  fixed 
and  B   moving,  or  we  may  think  of  B   as   fixed   and  A   in   motion.     Both 
methods  are  correct,  and  each  is  capable  of  giving  an  accurate  description  of 
the  change  in  relative  distance  and  direction.    So,  in  the  case  of  the  celestial 
bodies,   we    may   describe   the   variation   in    their    distances    and    directions, 
either  by  assuming  the  earth  to  be  fixed  with  the  remaining  bodies  in  motion, 
or  by  choosing  another  body,  the  sun,  as  the  fixed  member  of  the  system 
and  describing  the  phenomena  in  terms  of  motions  referred  to  it.    The  former 
method  of  procedure  is  the  starting  point  for  the  system  of  Ptolemy,  the 
latter,  for  that  of  Copernicus.     Both  methods  are  correct,  and  hence  neither 
can  give  rise  to  contradiction  so  long  as  the  problem  remains  one  of  motion. 
Though  two  ways  lie  open  before  us,  both  leading  to  the  same  goal,  the 
choice  of  route  is  by  no  means  a  matter  of  indifference,  for  one  is  much  more 
direct  than  the  other.     For  the  discussion  of  many  questions  the  conception 
of  a  fixed  earth  and  rotating  heavens  affords  a  simpler  method  of  treatment ; 
but,  when  a  detailed  description  of  the  motions  of  the  planets  and  satellites 
is  required,  the  Copernican  system  is  the  more  useful  by  far,  although  the 
geocentric  theory  presents  no  formal  contradiction  unless  we  pass  beyond  the 
consideration  of  the  phenomena  as  a  case  of  relative  motion,  and  attempt 
their  explanation  as  the  result  of  the  action  of  forces  and  accelerations.     If 
this  be  done,  the  conception  which  makes  the  earth  the  central  body  of  the 
universe  comes  into  open  conflict  with  the  fundamental  principles  of  me- 
chanics.    With  the  heliocentric  theory  there  is  no  such  conflict,  and  herein 
lies  the  essence  of  the  various   so-called   proofs   of  the   correctness   of  the 
Copernican  system. 

The  problems  of  practical   astronomy   are   among  those  which   can   1 
more  simply  treated  on  the  basis  of  the  geocent'ric  theory,  and  we  might 


INTRODUCTION  7 

have  proceeded  to  an  immediate  consideration  of  our  subject  from  this 
primitive  stand-point  but  for  the  importance  of  emphasizing  the  character 
of  what  we  are  about  to  do.  For  the  sake  of  simplicity,  we  shall  make  use 
of  ideas  which  are  not  universally  applicable  throughout  the  science  of  as- 
tronomy. We  shall  speak  of  a  fixed  earth  and  rotating  heavens  because  it 
is  convenient,  and  for  our  present  purpose,  precise ;  but,  in  so  doing,  it  is  im- 
portant always  to  bear  in  mind  the  more  elaborate  scheme  outlined  above, 
and  be  ever  ready  to  shift  our  view-point  from  the  relatively  simple,  elemen- 
tary conceptions  which  form  a  part  of  our  daily  experience,  to  the  more  ma- 
jestic structure  whose  proportions  and  dimensions  must  ever  be  the  delight 
and  wonder  of  the  human  mind. 

4.  Relation  of  the  problems  of  practical  astronomy  to  the  phenomena 
of  the  heavens. — The  problems  of  practical  astronomy  with  which  we  are 
concerned  are  the  determination  of  latitude,  azimuth,  time,  and  longitude. 

(a)  The  latitude  of  a  point  on  the  earth  may  be  defined  roughly  as  its 
angular  distance  from  the  equator.     It  can  be  shown  that  this  is  equal  to 
the  complement  of  the  inclination  of  the  rotation  axis  of  the  celestial  sphere 
to  the  direction  of  the  plumb  line  at  the  point  considered.     If  the  inclination 
of  the  axis  to  the  plumb  line  can  be  determined,  the  latitude  at  once  becomes 
known. 

(b)  The  azimuth  of  a  point  is  the  angle  included  between  the  vertical 
plane  containing  the  rotation  axis  of  the  celestial  sphere  and  the  vertical 
plane  through  the  object.     If  the  orientation  of  the  vertical  plane  through 
the  axis  of  the  sphere  can  be  found,  the  determination  of  the  azimuth  of  the 
point  becomes  but  a  matter  of  instrumental  manipulation. 

(c)  Time  measurement  is  based  upon  the  diurnal  rotation  of  the  earth, 
which  appears  to   us   in  reflection   as   the   diurnal   rotation   of  the   celestial 
sphere.    The  rotation  of  the  celestial  sphere  can  therefore  be  made  the  basis 
of   time    measurement.     To    determine   the    time   at    any    instant,   we    have 
only  to  find  the  angle  through  which  the   sphere  has   rotated   since  some 
specified  initial  epoch. 

(d)  As  will  be  seen  later,  the  determination  of  the  difference  in  longi- 
tude of  two  points  is  equivalent  to  finding  the  difference  of  their  local  times. 
The  solution  of  the  longitude  problem  therefore  involves  the  application  of 
the  methods  used  for  the  derivation  of  time,  together  with  some  means;  of 
comparing  the  local  times  of  the  two  places.    The  latter  can  be  accomplished 
by   purely    mechanical    means,    quite     independently     of     any     astronomical 
phenomena,   although  such   phenomena   are  occasionally   used   for  the   pur- 
pose. 

In  brief,  therefore,  the  solution  of  these  four  fundamental  problems 
can  be  connected  directly  with  certain  fundamental  celestial  phenomena. 
Both  latitude  and  azimuth  depend  upon  the  position  of  the  rotation  axis  of 
the  celestial  sphere,  the  former,  upon  its  inclination  to  the  direction  of  the 
plumb  line,  the  latter,  upon  the  orientation  of  the  vertical  plane  passing 
through  it ;  while  the  determination  of  time  and  longitude  involve  the  posi- 
tion of  the  sphere  as  affected  by  diurnal  rotation. 


8  PRACTICAL  ASTRONOMY 

A  word  more,  and  we  are  immediately  led  to  the  detailed  consideration 
of  our  subject:  The  solution  of  our  problems  requires  a  knowledge  of  the 
position  of  the  axis  of  the  celestial  sphere  and  of  the  orientation  of  the 
sphere  about  that  axis.  W'e  meet  at  the  outset  a  difficulty  in  that  the  sphere 
and  its  axis  have  no  objective  existence.  Since  our  observations  and  meas- 
urements must  be  upon  things  which  have  visible  existence,  the  stars  for 
example,  we  are  forced  to  an  indirect  method  of  procedure.  We  must  make 
our  measurements  upon  the  various  celestial  bodies  and  then,  from  the 
known  location  of  these  objects  on  the  sphere,  derive  the  position  of  the 
sphere  and  its  axis.  This  raises  at  once  the  general  question  of  coordinates 
and  coordinate  systems  to  which  we  now  give  our  attention. 

5.  Coordinates  and  Coordinate  Systems. — Position  is  a  relative  term. 
We  cannot  specify  the  position  of  any  object  without  referring  it,  either 
explicitly  or  implicitly,  to  some  other  object  whose  location  is  assumed  to 
be  known.  The  designation  of  the  position  of  a  point  on  the  surface  of 
a  sphere  is  most  conveniently  accomplished  by  a  reference  to  two  great 
circles  that  intersect  at  right  angles.  For  example,  the  position  of  a  point 
on  the  earth  is  fixed  by  referring  it  to  the  equator  and  some  meridian 
as  that  of  Greenwich  or  Washington.  The  angular  distance  of  the  point 
from  the  circles  of  reference  are  its  coordinates — in  this  case,  longitude  and 
latitude. 

Our  first  step,  therefore,  in  the  establishment  of  coordinate  systems  for 
the  celestial  sphere,  is  the  definition  of  the  points  and  circles  of  reference 
which  will  form  the  foundation  for  the  various  systems. 

The  Direction  of  the  Plumb  Line,  or  the  Direction  due  to  Gravity, 
produced  indefinitely  in  both  directions,  pierces  the  celestial  sphere  above  in 
the  Zenith,  and  below,  in  the  Nadir.  The  plane  through  the  point  of  obser- 
vation, perpendicular  to  the  direction  of  the  plumb  line,  is  called  the  Hor- 
izon Plane.  Produced  indefinitely  in  all  directions,  it  cuts  the  celestial 
sphere  in  a  great  circle  called  the  Horizon.  Since  the  radius  of  the  celestial 
sphere  is  indefinitely  great  as  compared  with  the  radius  of  the  earth,  a 
plane  through  the  center  of  the  earth  perpendicular  to  the  direction  of 
gravity  will  also  cut  the  celestial  sphere  in  the  horizon.  For  many  pur- 
poses it  is  more  convenient  to  consider  this  plane  as  the  horizon  plane. 

The  celestial  sphere  is  pierced  by  its  axis  of  rotation  in  two  points 
called  the  North  Celestial  Pole  and  the  South  Celestial  Pole,  or  more  briefly, 
the  North  Pole  and  the  South  Pole,  respectively.  It  is  evident  from  the 
relations  between  the  phenomena  and  their  interpretation  traced  in  Section 
3  that  the  axis  of  the  celestial  sphere  must  coincide  with  the  earth's  axis  of 
rotation. 

Great  circles  through  the  zenith  and  nadir  are  called  Vertical  Circles. 
Their  planes  are  perpendicular  to  the  horizon  plane.  The  vertical  circle 
passing  through  the  celestial  poles  is  called  the  Celestial  Meridian,  or  simply, 
the  Meridian.  Its  plane  coincides  with  the  plane  of  the  terrestrial  meridian 
through  the  point  of  observation. 


DEFINITIONS 

The  vertical  circle  intersecting-  the  meridian  at  an  angle  of  ninety  degrees 
is  called  the  Prime  Vertical.  The  intersections  of  the  meridian  and  prime 
vertical  with  the  horizon  are  the  cardinal  points,  North,  East,  South,  and 
West. 

Small  circles  parallel  to  the  horizon  are  called  Circles  of  Altitude  or 
Almucanters. 

Great  circles  through  'the  poles  of  the  celestial  sphere  are  called  Hour 
Circles. 

The  great  circle  equatorial  to  the  poles  of  the  celestial  sphere  is  called 
the  Celestial  Equator.  The  plane  of  the  celestial  equator  coincides  with 
the  plane  of  the  terrestrial  equator. 

Small  circles  parallel  to  the  celestial  equator  are  called  Circles  of  Dec- 
lination. 

The  ecliptic,  already  defined  as  the  great  circle  of  the  celestial  sphere  fol- 
lowed by  the  sun  in  its  annual  motion  among  the  stars,  is  inclined  to  the 
celestial  equator  at  an  angle  of  about  23^  degrees.  The  points  of  inter- 
section of  the  ecliptic  and  the  celestial  equator  are  the  Equinoxes,  Vernal 
and  Autumnal,  respectively.  The  Vernal  Equinox  is  that  point  at  which 
the  sun  in  its  annual  motion  passes  from  the  south  to  the  north  side  of  the 
equator;  the  Autumnal  Equinox,  that  at  which  it  passes  from  the  north  to 
the  south. 

The  points  on  the  ecliptic  midway  between  the  equinoxes  are  called  the 
Solstices,  Summer  and  Winter,  respectively.  The  Summer  Solstice  lies  to 
the  north  of  the  celestial  equator,  the  Winter  Solstice,  to  the  south. 

The  coordinate  systems  most  frequently  used  in  astronomy  present 
certain  features  in  common,  and  a  clear  understanding-  of  the  underlying 
principles  will  greatly  aid  in  acquiring  a  knowledge  of  the  various  systems. 
At  the  basis  of  each  system  is  a  Fundamental  Great  Circle.  Great  circles 
perpendicular  to  this  are  called  Secondary  Circles.  One  of  these,  called  the 
Principal  Secondary,  and  the  fundamental  great  circle,  form  the  reference 
circles  of  the  system. 

The  Primary  Coordinate  is  measured  along  the  fundamental  great  circle 
from  the  principal  secondary  to  the  secondary  passing  through  the  object 
to  which  the  coordinates  refer.  The  Secondary  Coordinate  is  measured  along 
the  secondary  passing  through  the  object  from  the  fundamental  great  circle 
to  the  object  itself.  The  fundamental  great  circle  and  the  principal  secondary 
intersect  in  two  points.  The  intersection  from  which  the  primary  coordinate 
is  measured,  and  the  direction  of  measurement  of  both  coordinates,  must  be 
specified. 

In  practical  astronomy  three  systems  of  coordinates -are  required.  The 
details  are  shown  by  the  following  table.  The  symbol  used  to  designate 
each  coordinate  is  written  after  its  name  in  the  table. 

It  is  sometimes  more  convenient  to  use  as  secondary  coordinate  the  dis- 
tance of  the  object  from  one  of  the  poles  of  the  fundamental  great  circle. 
Thus  in  System  I  we  shall  frequently  use  the  distance  of  an  object  from 


10 


PR  A  C  TIC  A  L  A  S  TR  ONOM1 ' 


COORDINATE  SYSTEMS. 


SYSTEM 

FUNDAMENTAL 
GREAT  CIRCLE 

SECONDARY 
CIRCLES 

PRINCIPAL 
SECONDARY 

COORDINATES 

PRIMARY 

SECONDARY 

I 

Horizon 

Vertical 

'Meridian 

Azimuth  =  A 

Altitude  =  h 

Circles 

-j-  from  South 
toward  West 

-\-  from  Horizon 
upward 

II 

Celestial 
Equator 

Hour 
Circles 

Hour  Circle 
coinciding 
with 
Meridian 

Hour  Angle  —  t 
-f-  from  Meridian 
toward  West 

Declination  =  $ 
-f-  from  Equator 
toward  North; 
—  toward  South 

III 

« 

u 

Hour  Circle 
through   the 
Vernal 
Equinox 

Right  Ascension 
=  a     +  from 
Vernal    Equinox 
toward  East 

11            it 

the  zenith,  its  Zenith  Distance  =  s,  instead  of  the  altitude.  Similarly,  in 
Systems  II  and  III  we  shall  occasionally  find  that  an  object's  distance  from 
the  north  celestial  pole,  its  North  Polar  Distance  —  -,  is  more  convenient  than 
declination.  Between  these  alternative  coordinates  we  have  the  relations : 


(0 


r:  =  90°  —  o 


The  details  of  the  various  systems  are  also  shown  graphically  in  Fig.  1, 
which  represents  an  orthogonal  projection  of  the  celestial  sphere  upon  the 
horizon  plane.  In  this  projection  all  vertical  circles  become  straight  lines. 
All  circles  inclined  to  the  horizon  at  an  angle  other  than  90°  become  ellipses. 
The  horizon,  and  all  circles  parallel  to  the  horizon  plane,  remain  circles. 

6.  Characteristics  of  the  Three  Systems.  Changes  in  the  Coordinates. — 
Coordinates  are  used  both  for  the  location  of  objects  on  the  sphere  by  actual 
observation,  and  as  a  means  of  stating  positions  predicted  on  the  basis  of 
the  laws  which  describe  the  motions  of  the  various  celestial  bodies.  The 
practical  astronomer  and  the  engineer  have  occasion  to  use  them  in  both 
ways.  It  is  essential  that  there  be  a  clear  understanding  of  the  relative 
advantages  of  the  various  systems,  of  the  changes  which  may  occur  in  the 
different  coordinates,  and  of  the  relations  of  the  systems  to  each  other. 
We  now  proceed  to  a  discussion  of  the  first  two  of  these  points.  The  rela- 
tions between  the  systems  will  be  discussed  in  Chapter  II. 

None  of  the  coordinates  defined  above  is  absolutely  constant  for  any 
of  the  celestial  bodies.  The  changes  which  occur  arise  as  the  result  of: 

(a)  a  change  in  the    position    of   the   object, 

(b)  a  change  in  the  position  of  the  reference  circles, 

(c)  a  change  in  the  position  of  the  observer, 

(d)  a  bending  of  the  light  rays  by  the  atmosphere  surrounding  the  earth. 


COORDINATES 


11 


Fig.  i. 


Point  Z 

Circle  NESW 

Point  P 

Line  NZS 

Line  WZE 

Points  .V,  E,  S,  W, 

Ellipse  WME 
Ellipse 
Point 
Point 


Line 
Arc 


Arc  SI- 


TV  m  e 
V 
O 

Z  Or 
POp 

Angle     SZO 


Line 
Line 

Arc  Mp  —  Angle 
Arc 
Arc 

Arc 


rO 

ZO 

ZPO 
pO 
PO 

vp 


Zenith 

Horizon 

North  Celestial  Pole 

Celestial  Meridian 

Prime  Vertical 

Cardinal  Points 

Celestial  Equator 

Ecliptic 

Vernal  Equinox 

Any  Celestial  Object 

Vertical  Circle  through  O 

Hour  Circle  through  O 

Azimuth  ot  O  =  A 
Altitude  of  6  =  h 
Zenith  Distance  of  O  =  z 

Hour  Angle  of  O  =  t 
Declination  of  O  —  8 
North  Polar  Distance  of  O  = 

Right  Ascension  of  O  =  « 
,7  and  -  same  as  in  System  II 


I  Coordinates 
f  System  I 

I  Coordinates 
\  System  II 

~|  Coordinates 
f  System  III 


Any  or  all  of  these  causes  may  enter  to  affect  the  position  of  an  object, 
with  the  result  that  the  number  of  possible  variations  with  which  we  have 
to  deal  is  considerable.  In  some  instances,  however,  the  variations  are  small 
quantities — so  small  that  they  can  be  disregarded  in  all  but  the  most  precise 
investigations.  The  small  changes  which  cannot  be  neglected  entirely  are 
regarded  as  corrections,  which,  applied  to  the  coordinates  corresponding  to 
a  given  position  of  the  object,  reference  circles,  and  observer,  give  their  values 
for  some  other  position. 


12  PRACTICAL  ASTRONOMY 

The  bending  of  the  light  rays  by  the  earth's  atmosphere,  a  phenomenon 
known  as  Refraction,  affects  all  of  the  coordinates  but  azimuth.1  The 
amount  of  the  refraction,  which  is  always  small,  depends  upon  the  conditions 
under  which  the  object  is  observed.  The  allowance  for  its  influence  is  there- 
fore made  by  each  individual  observer.  The  method  of  determining  its 
amount  will  be  discussed  in  Section  8. 

In  the  first  system,  the  reference  circles  are  fixed  for  any  given  point 
of  observation.  The  azimuth  and  altitude  of  terrestrial  objects  are  therefore 
constant,  unless  the  point  of  observation  is  shifted.  For  celestial  bodies,  on 
the  contrary,  they  are  continuously  varying.  The  positions  of  all  such  objects 
are  rapidly  and  constantly  changing  with  respect  to  the  circles  of  reference, 
as  a  result  of  the  diurnal  rotation.  For  the  nearer  bodies,  an  additional 
complexity  is  introduced  by  their  motions  over  the  sphere  and  the  changing 
position  of  the  earth  in  its  orbit.  It  appears,  therefore,  that  azimuth  and 
altitude  are  of  special  service  in  surveying  and  in  geodetic  operations,  but 
that  their  range  of  advantageous  application  in  connection  with  celestial 
bodies  is  limited,  for  not  only  are  the  azimuth  and  altitude  of  a  celestial  object 
constantly  changing,  but,  for  any  given  instant,  their  values  are  different 
for  all  points  on  the  earth.  But  in  spite  of  this  disadvantage,  altitude,  at  least, 
is  of  great  importance.  Its  determination  in  the  case  of  a  celestial  body 
affords  convenient  methods  of  solving  two  of  the  fundamental  problems  with 
which  we  are  concerned,  viz.,  latitude  and  time.  Since  the  fundamental 
circle  in  the  first  system  depends  only  upon  the  direction  of  the  plumb 
line,  the  instrument  required  for  the  measurment  of  altitude  is  extremely 
simple,  both  in  construction  and  use.  In  consequence,  altitude  is  the  most 
readily  determined  of  all  the  various  coordinates.  The  observational  part 
of  the  determination  of  latitude  and  time  is  therefore  frequently  based  upon 
measures  of  altitude,  the  final  results  being  derived  from  the  observed  data 
by  a  process  of. coordinate  transformation  to  be  developed  in  Chapter  II. 

In  the  third  system,  the  reference  circles  share  in  the  diurnal  rotation. 
Although  not  absolutely  fixed  on  the  sphere,  their  motions  are  so  slow  that 
the  coordinates  of  objects,  which,  like  the  stars,  are  sensibly  fixed,  remain 
practically  constant  for  considerable  intervals  of  time.  Right  ascension  and 
declination  are  therefore  convenient  for  listing  or  cataloguing  the  positions 
of  the  stars.  Catalogues  of  this  sort  are  not  only  serviceable  for  long  periods 
of  time,  but  can  also  be  used  at  all  points  on  the  earth.  The  latter  circum- 
stance renders  right  ascension  and  declination  an  advantageous  means  of 
expressing  the  positions  of  bodies  not  fixed  on  the  sphere.  For  such  objects 
we  have  only  to  replace  the  single  pair  of  coordinates  which  suffices  for  a 
star,  by  a  series  giving  the  right  ascension  and  declination  for  equi-distant 
intervals  of  time.  Such  a  list  of  positions  is  called  an  Ephemeris.  If  the 
time  intervals  separating  the  successive  epochs  for  which  the  coordinates  are 
given  be  properly  chosen,  the  position  can  be  found  for  any  intermediate 

JThe  azimuth  of  objects  near  the  horizon  is  also  affected  by  refraction.  The  magnitude 
of  the  change  in  the  coordinate  is  very  small,  however. 


COORDINATES  13 

instant  by  a  process  of  interpolation.  The  interval  selected  for  the  tabula- 
tion is  determined  by  the  rapidity  and  regularity  with  which  the  coordinates 
change.  In  the  case  of  the  sun,  one  day  intervals  are  sufficient,  but  for  the 
moon  the  positions  must  be  given  for  each  hour.  For  the  more  distant 
planets,  whose  motions  are  relatively  slow,  the  interval  can  be  increased 
to  several  days. 

Collections  of  ephemerides  of  the  sun,  moon,  and  the  planets,  together 
with  the  right  ascensions  and  declinations  of  the  brighter  stars,  are  pub- 
lished annually  by  the  governments  of  the  more  important  nations.  That 
issued  by  our  own  is  prepared  in  the  Nautical  Almanac  Office  at  Washington, 
and  bears  the  title  "American  Ephemeris  and  Nautical  Almanac." 

It  is  necessary  to  examine  the  character  of  the  variations  produced 
in  the  coordinates  by  the  slow  motion  of  the  reference  circles  mentioned 
above.  The  mutual  attractions  of  the  sun,  moon,  and  the  planets  produce 
small  changes  in  the  positions  of  the  equator  and  ecliptic.  The  motion  of 
the  ecliptic  is  relatively  unimportant.  That  of  the  equator  is  best  understood 
by  tracing  the  changes  in  position  of  the  earth's  axis  of  rotation.  As  the 
earth  moves  in  its  orbit,  the  axis  does  not  remain  absolutely  parallel  to  a 
given  initial  position,  but  describes  a  conical  surface.  The  change  in  the 
direction  of  the  axis  takes  place  very  slowly,  about  26000  years  being  required 
for  it  to  return  to  its  original  position.  During  this  interval  the  inclination 
of  the  equator  to  the  ecliptic  never  deviates  greatly  from  its  mean  value 
of  about  23^°.  Consequently,  the  celestial  pole  appears  to  move  over 
the  sphere  in  a  path  closely  approximating  a  circle  with  the  pole  of  the 
ecliptic  as  center.  The  direction  of  the  motion  is  counter-clockwise,  and 
the  radius  of  the  circle  equal  to  the  inclination  of  the  equator  to  the  ecliptic. 
The  actual  motion  of  the  pole  is  very  complex;  but  its  characteristic  features 
are  the  progressive  circular  component  already  mentioned,  and  a  transverse 
component  which  causes  it  to  oscillate  or  nod  back  and  forth  with  respect 
to  the  pole  of  the  ecliptic.  The  result  is  a  vibratory  motion  of  the  equator 
about  a  mean  position  called  the  Mean  Equator,  the  mean  equator  itself 
slowly  revolving  about  a  line  perpendicular  to  the  plane  of  the  ecliptic. 
The  motion  of  the  equator  combined  with  that  of  the  ecliptic  produces  an 
oscillation  of  the  equinox  about  a  mean  position  called  the  Mean  Vernal 
Equinox,  which,  in  turn,  has  a  slow  progressive  motion  toward  the  west. 
The  resulting  changes  in  the  right  ascension  and  declination  are  divided  into 
two  classes,  called  precession  and  nutation,  respectively.  Precession  is  that 
part  of  the  change  in  the  coordinates  arising  from  the  progressive  westward 
motion  of  the  mean  vernal  equinox,  while  Nutation  is  the  result  of  the 
oscillatory  or  periodic  motion  of  the  true  vernal  equinox  with  respect  to  the 
mean  equinox. 

The  amount  of  the  precession  and  nutation  depends  upon  the  position 
of  the  star.  For  an  object  on  the  equator  the  maximum  value  of  the  preces- 
sion in  right  ascension  for  one  year  is  about  forty-five  seconds  of  arc  or  three 
seconds  of  time.  For  stars  near  the  pole  it  is  much  larger,  amounting  in 


14  PRACTICAL  ASTRONOMY 

the  case  of  Polaris,  for  example,  to  about  2'7S.  The  annual  precession 
in  declination  is  relatively  small,  and  does  not  exceed  20"  for  any  of  the 
stars. 

There  remains  to  be  considered  the  effect  of  the  object's  own  motion 
and  that  of  the  observer.  We  ha\re  already  seen  how  the  changes  arising 
from  the  motion  of  such  objects  as  the  sun,  moon,  and  the  planets  can  be 
expressed  by  means  of  an  ephemeris  giving  the  right  ascension  and  declina- 
tion for  equi-distant  intervals  of  time.  For  the  stars  the  matter  is  much 
simpler.  Their  motions  over  the  sphere  are  so  slight  as  to  be  entirely  inap- 
preciable in  the  vast  majority  of  cases,  and  for  those  in  which  the  change 
cannot  be  disregarded,  it  is  possible  to  assume  that  the  motion  is  uniform 
and  along  the  arc  of  a  great  circle.  The  change  in  one  year  is  called  the 
star's  Proper  Motion.  If  the  right  ascension  and  declination  are  given  for 
any  instant,  /,  and  it  is  desired  to  find  their  values  as  affected  by  proper 
motion  for  any  other  instant  /',  it  is  only  necessary  to  add  to  the  given 
coordinates  the  products  of  the  proper  motion  in  right  ascension  and  declina- 
tion into  the  interval/  —  f  expressed  in  years.  The  position  of  a  star  for  a 
given  initial  epoch  and  its  proper  motion  are  therefore  all  that  is  required 
for  the  determination  of  its  position  at  any  other  epoch,  in  so  far  as  the 
position  is  dependent  upon  the  star's  own  motion. 

The  motion  of  the  observer  may  affect  the  position  of  a  celestial  object 
in  two  ways :  First,  the  actual  change  in  his  position  due  to  the  diurnal 
and  annual  motions  of  the  earth  causes  a  change  in  the  coordinates  called 
Parallactic  Displacement.  Second,  the  fact  that  the  observer  is  in  motion 
at  the  instant  of  observation  may  produce  an  apparent  change  in  the  direction 
in  which  the  object  is  seen,  in  the  same  way  that  the  direction  of  the  wind, 
as  noted  from  a  moving  boat  or  train,  appears  different  from  that  when  the 
observer  is  at  rest.  The  change  thus  produced  is  called  Aberration,  and  is 
carefully  to  be  distinguished  from  the  parallactic  displacement.  Aberration 
depends  only  upon  the  observer's  velocity,  and  not  at  all  upon  his  position, 
except  as  position  may  determine  the  direction  and  magnitude  of  the  motion. 
Parallactic  displacement,  on  the  contrary,  depends  on  the  distance  over 
which  the  observer  actually  moves. 

For  the  nearer  bodies  the  parallactic  displacement  due  to  the  earth's 
annual  motion  is  large,  and  is  included  with  the  effect  of  the  object's  own 
motion  in  the  ephemeris  which  expresses  its  positions.  The  variation  arising 
from  the  rotation  of  the  earth  on  its  axis  is  far  smaller,  and  can  always 
be  treated  as  a  correction.  In  the  case  of  the  stars,  the  distances  are  so 
great  that  the  maximum  known  parallactic  displacement  due  to  the  earth's 
annual  motion  amounts  to  only  three-quarters  of  a  second  of  arc.  For  all 
but  a  few,  a  shift  in  the  position  of  the  earth  from  one  side  of  its  orbit  to 
the  other,  a  distance  of  more  than  180,04)0,000  miles,  reveals  no  measurable 
change  in  the  coordinates.  The  displacement  due  to  the  earth's  rotation  is 
of  course  altogether  inappreciable. 

Parallactic  displacement  is  usually  called  Parallax,  and,  when  so  spoken 
of,  signifies  specifically,  the  correction  which  must  be  applied  to  the  observed 


15 

coordinates  of  an  object  in  order  to  reduce  them  to  what  they  would  be 
were  the  object  seen  from  a  standard  position.  For  the  stars,  the  standard 
position  is  the  center  of  the  sun ;  for  all  other  bodies,  the  center  of  the  earth. 

Aberration  is  due  to  the  fact  that  the  velocity  of  the  observer  is  a  quantity 
of  appreciable  magnitude  as  compared  with  the  velocity  of  light.  For  all 
stars  not  lying  in  the  direction  of  the  earth's  orbital  motion,  the  telescope 
must  be  inclined  slightly  in  advance  of  the  star's  real  position  in  order  that 
its  rays  may  pass  centrally  through  both  objective  and  eye-piece  of  the 
instrument.  The  star  thus  appears  displaced  in  the  direction  of  the  ob- 
server's motion.  The  amount  of  the  displacement  is  a  maximum  when  the 
direction  of  the  motion  is  at  right  angles  to  the  direction  of  the  star,  and 
equal  to  zero  when  the  two  directions  coincide.  The  rotation  of  the  earth 
on  its  axis  produces  a  similar  displacement.  The  Diurnal  Aberration  is  so 
minute,  however,  that  it  requires  consideration  only  in  the  most  refined 
observations. 

The  coordinates  of  the  second  system  possess,  to  a  certain  degree,  the 
properties  of  those  of  both  Systems  I  and,  III.  Hour  angle,  like  azimuth 
and  altitude,  is  a  coordinate  which  varies  continuously  and  rapidly,  and  is 
dependent  on  the  position  of  the  observer  on  the  earth.  The  secondary 
coordinate,  declination,  is  the  same  as  in  System  III,  and  the  remarks  con- 
cerning it  made  above,  apply  with  equal  force  here.  The  second  system 
is  of  prime  importance  in  the  solution  of  the  problems  of  practical  astronomy, 
for  it  serves  as  an  intermediate  step  in  passing  from  System  I  to  System  III, 
or  vice  versa.  It  is  also  the  basis  for  the  construction  of  the  equatorial 
mounting  for  telescopes,  the  form  most  commonly  used  in  astronomical  inves- 
tigations. 

7.  Summary.  Method  of  treating  the  corrections  in  practice. — It  is 
to  be  remembered,  therefore,  that  the  azimuth  and  altitude  of  terrestrial 
objects  are  constant  for  a  given  point  of  observation,  but  change  as  the 
observer  moves  over  the  surface  of  the  earth.  For  celestial  objects  they 
are  not  only  different  for  each  successive  instant,  but  also,  for  the  same 
instant,  they  are  different  for  different  points  of  observation.  Right  ascension 
and  declination  are  sensibly  the  same  for  all  points  on  the  earth,  and,  in  con- 
sequence, are  used  in  the  construction  of  catalogues  and  ephemerides.  One 
pair  of  values  serves  to  fix  the  position  of  a  star  for  a  long  period  of  time, 
but  for  the  sun,  the  moon,  and  the  planets  an  ephemeris  is  required. 

The  corrections  to  which  the  coordinates  are  subject  are  proper  motion, 
precession,  nutation,  annual  aberration,  diurnal  aberration,  parallax,  stellar 
or  planetary  as  the  case  may  be,  and  refraction.  Right  ascension  and  decli- 
nation are  affected  by  all,  but  only  planetary  parallax,  refraction,  and  diurnal 
aberration  arise  in  practice  in  connection  with  azimuth  and  altitude,  and 
of  these  three  the  last  is  usually  negligible.  In  all  cases  these  three  are 
dependent  upon  local  conditions,  and  consequently,  their  calculation  and 
application  are  left  to  the  observer.  Since  it  is  impracticable  to  include 
them  in  catalogue  and  ephemeris  positions  of  right  ascension  and  declination, 
there  remains  to  be  considered,  as  affecting-  such  positions,  proper  motion. 


16  PRACTICAL  ASTRONOMT 

precession,  nutation,  annual  aberration,  and  stellar  parallax.  The  last  is  so 
rarely  of  significance  in  practical  astronomy  that  it  can  be  disregarded.  As 
for  the  others,  it  is  sometimes  necessary  to  know;  their  collective  effect,  and 
sometimes,  the  influence  of  the  individual  variations.  It  thus  happens 
that  we  have  different  kinds  of  positions  or  places,  known  as  mean  place, 
true  place,  and  apparent  place. 

The  mean  place  of  an  object  at  any  instant  is  its  position  referred  to 
the  mean  equator  and  mean  equinox  of  that  instant.  The  mean  place  is 
affected  by  proper  motion  and  precession. 

The  true  place  of  an  object  at  any  instant  is  its  position  referred  to  the 
true  equator  and  true  equinox  of  that  instant,  that  is,  to  the  instantaneous 
positions  of  the  actual  equator  and  equinox.  The  true  place  is  equal  to  the 
mean  place  plus  the  variation  due  to  the  nutation. 

The  apparent  place  of  an  object  at  any  instant  is  equal  to  the  true  place 
at  that  instant  plus  the  effect  of  annual  aberration.  It  expresses  the  location 
of  the  object  as  it  would  appear  to  an  observer  situated  at  the  center  of 
the  earth. 

The  positions  to  be  found  in  star  catalogues  are  mean  places,  and  are 
referred  to  the  mean  equator  and  equinox  for  the  beginning  of  some  year, 
for  example,  1855.0  or  1900.0.  Such  catalogues  usually  contain  the  data  neces- 
sary for  the  determination  of  the  precession  corrections  which  must  be  applied 
to  the  coordinates  in  deriving  the  mean  place  for  any  other  epoch.  Modern 
catalogues  also  contain  the  value  of  the  proper  motion  when  appreciable. 
The  nutation  and  annual  aberration  corrections  are  found  from  data  given 
by  the  various  annual  ephemerides.  The  ephemerides  themselves  contain 
mean  places  for  several  hundred  of  the  brighter  stars ;  but  the  engineer  is 
rarely  concerned  with  these,  or  with  the  catalogue  positions  mentioned 
above,  for  apparent  places  are  also  given  for  the  ephemeris  stars,  and  these 
are  all  that  he  needs.  The  apparent  right  ascension  and  declination  are 
given  for  each  star  for  every  ten  days  throughout  the  year.  Apparent  posi- 
tions are  also  given  by  the  ephemeris  for  the  sun,  the  moon,  and  the  planets, 
for  suitably  chosen  intervals.  Positions  for  all  of  these  bodies  for  dates 
intermediate  to  the  ephemeris  epochs  can  be  found  by  interpolation.  With 
this  arrangement,  the  special  calculation  of  the  various  corrections  necessary 
for  the  formation  of  apparent  places  is  avoided  entirely  in  the  discussion  of 
all  ordinary  observations.  The  observer  must  understand  the  origin  and 
significance  of  all  of  the  changes  which  occur  in  the  coordinates,  in  order 
to  use  the  ephemeris  intelligently ;  but  he  has  occasion  to  calculate  specially 
only  those  which  depend  upon  the  local  conditions  affecting  the  observations, 
viz.,  diurnal  aberration,  parallax,  and  refraction.  The  first  we  disregard  on 
account  of  its  minuteness.  There  remains  for  the  consideration  of  the  engi- 
neer only  refraction  and  parallax.  The  following  is  a  brief  statement  of 
the  methods  by  which  their  numerical  values  can  be  derived. 

8.  Refraction. — The  velocity  of  light  depends  upon  the  density  of  the 
medium  which  it  traverses.  When  a  luminous  disturbance  passes  from  a 
medium  of  one  density  into  that  of  another,  the  resulting  change  in  velocity 


REFRACTION  17 

shifts  the  direction  of  the  wave  front,  unless  the  direction  of  propagation  is 
perpendicular  to  the  surface  separating  the  two  media.  Stated  otherwise, 
a  light  ray  passing  from  one  medium  into  another  of  different  density  under- 
goes a  change  in  direction,  unless  the  direction  of  incidence  is  normal  to 
the  bounding  surface.  This  change  in  direction  is  called  Refraction.  The 
incident  ray,  the  refracted  ray,  and  the  normal  to  the  bounding  surface  at 
the  point  of  incidence  lie  in  a  plane.  When  the  density  of  the  second  medium 
is  greater  than  that  of  the  first,  the  ray  is  bent  toward  the  normal.  When 
the  conditions  of  density  are  reversed,  the  direction  of  bending  is  away  from 
the  normal. 

The  light  rays  from  a  celestial  object  which  reach  the  eye  of  the  observer 
must  penetrate  the  atmosphere  surrounding  the  earth.  They  pass  from  a 
region  of  zero  density  into  one  whose  density  gradually  increases  from  the 
smallest  conceivable  amount  to  a  maximum  which  occurs  at  the  surface  of 
the  earth.  The  rays  undergo  a  change  in  direction  as  indicated  above.  The 
effect  is  to  increase  the  altitude  of  all  celestial  bodies,  without  sensibly 
changing  their  azimuth  unless  they  are  very  near  the  horizon.  For  the 
case  of  two  media  of  homogeneous  density,  the  phenomenon  of  refraction 
is  simple;  but  here,  it  is  extremely  complex  and  its  amount  difficult  of 
determination.  The  course  of  the  ray  which  reaches  the  observer  is  affected 
not  only  by  its  initial  direction,  but  also  by  the  refraction  which  it  suffers 
at  each  successive  point  in  its  path  through  the  atmosphere.  The  latter  is 
determined  by  the  density  of  the  different  strata,  which,  in  turn,  is  a  function 
of  the  altitude.  This  brings  us  to  the  most  serious  difficulty  in  the  problem, 
for  our  knowledge  of  the  constitution  of  the  atmosphere,  especially  in  its 
upper  regions,  is  imperfect.  To  proceed,  an  assumption  must  be  made  con- 
cerning the  nature  of  the  relation  connecting  density  and  altitude.  This, 
combined  with  the  fundamental  principles  enunciated  above,  forms  the  basis 
of  an  elaborate  mathematical  discussion  which  results  in  an  expression  giving 
the  refraction  as  a  function  of  the  zenith  distance  of  the  object,  and  the 
temperature  of  the  air  and  the  barometric  pressure  at  the  point  of  observa- 
tion. This  expression  is  complicated  and  cumbersome,  disadvantages  over- 
come, in  a  measure,  by  the  reduction  of  its  various  parts  to  tabular  form  in 
accordance  with  a  method  devised  by  Bessel.  With  this  arrangement,  the 
determination  of  the  refraction  involves  the  interpolation  and  combination 
of  a  half  dozen  logarithms,  more  or  less. 

Various  hypotheses  concerning  the  relation  between  density  and  altitude 
have  been  made,  each  of  which  gives  rise  to  a  distinct  theory  of  refraction, 
although  the  differences  between  the  corresponding  numerical  results  are 
slight.  That  generally  used  is  due  to  Gylden.  The  tables  based  upon  this 
theory  are  known  as  the  Pulkova  Refraction  Tables,  and  can  be  found  in  the 
more  comprehensive  works  on  spherical  and  practical  astronomy. 

When  the  highest  precision  is  desired  these  tables  or.  their  equivalent 
must  be  used,  but  for  many  purposes  a  simpler  procedure  will  suffice.  For 
example,  the  approximate  expression, 


PRACTICAL  ASTRONOMY 


983  ^    , 

r=  •*  J  .     tans  , 

460  +  / 


(3) 


derived  empirically  from  the  results  given  by  the  theoretical  development,1 
can  be  used  for  the  calculation  of"  the  refraction,  r,  .when  the  altitude  is  not 
less  than  15°.  In  this  expression,  b  is  the  barometer  reading  in  inches;  f, 
the  temperature  in  degrees  Fahrenheit ;  z,  the  observed  or  apparent  zenith 
distance.  The  refraction  is  given  in  seconds  of  arc.  The  error  of  the  result 
will  rarely  exceed  one  second. 

For  rough  work  the  matter  can  be  still  further  simplified  by  using  mean 
values  for  b  and  /.  For  £r=2<9.5  inches,  and  t  =r  50°  Fahr.  the  coefficient 
of  (3)  is  57",  whence 


=  57"  tan  z' . 


(4) 


The  values  of  r  given  by  (4)  can  be  derived  from  columns  three  and 
eight  of  Table  I  with  either  the  apparent  altitude  or  the  apparent  zenith 
distance  as  argument.  For  altitudes  greater  than  20°  and  normal  atmospheric 
conditions,  the  error  will  seldom  exceed  a  tenth  of  a  minute  of  arc. 

9.  Parallax. — The  parallax  of  an  object  is  equal  to  the  angle  at  the 
object  subtended  by  the  line  joining  the  center  of  the  earth  and  the  point 


Fig.    2 

of  observation.  Thus,  in  Fig.  2,  the  circle  represents  a  section  of  the  earth 
coinciding  with  the  vertical  plane  through  the  object.  C  is  the  center  of 
the  earth,  0  the  point  of  observation,  Z  the  zenith,  and  B  the  object.  The 
angles  z'  and  z  are  the  apparent  and  geocentric  zenith  distances,  respectively. 
Their  difference,  which  is  equal  to  the  angle  p,  is  the  parallax  of  B. 


form  was  derived    by  Comstock,   Bulletin  of  the    University  of   Wisconsin,  Science 
Series,  v.  i,  p.  60. 


PARALLAX  19 

We  have  the  relations 

z  =  z'—p,  (5) 

(6) 


where  h'  and  h  are  the  apparent  and  geocentric  altitudes,  respectively.  The 
effect  of  parallax,  therefore,  is  to  increase  zenith  distances  and  decrease 
altitudes,  —  just  the  opposite  of  that  produced  by  refraction. 

The  parallax  depends  upon  p  the  radius  of  the  earth,  r  the  distance 
of  the  object  from  the  earth's  center,  and  the  zenith  distance  z'  or  s.  From 
the  triangle  OCB 

/•sin/  =,0sin2'. 

The  angle  /»  does  not  exceed  a  few  seconds  of  arc  for  any  celestial  body 
excepting  the  moon.  For  this  its  maximum  value  is  about  1°.  We  therefore 
write 


p=smz'.  (7) 

The  coefficient  p  /  r,  the  value  of  the  parallax  when  the  body  is  the 
horizon,  is  called  the  Horizontal  Parallax.  Denoting  its  value  by/0  we  have 

/=/0sin^'.  (8) 

The  value  of  pQ  varies  with  the  distance  of  the  object.  It  is  tabulated 
in  the  American  Ephemeris  for  the  sun  (p.  285),  the  moon  (page  IV  of 
each  month),  and  the  planets  (pp.  218<-249).  For  the  sun,  however,  the 
change  in  /0  is  so  slight  that  we  may  use  its  mean  value  of  8"8,  whence 

p  =  8?8  sin  z.  (9) 

The  error  of  this  expression  never  exceeds  Q"3.  The  values  of  p  corre- 
sponding to  (9)  can  be  interpolated  from  columns  four  and  nine  of  Table  I. 

For  approximate  work  the  solar  parallax  is  conveniently  combined  with 
the  mean  refraction  given  by  (4).  The  difference  of  the  two  corrections 
can  be  derived  from  the  fifth  and  tenth  columns  of  Table  I  with  the  apparent 
altitude  or  the  apparent  zenith  distance  as  argument. 

The  preceding  discussion  assumes  that  the  earth  is  a  sphere.  On  this 
basis  the  parallax  in  azimuth  is  zero.  Actually,  the  earth  is  spheroidal 
in  form,  whence  it  results  that  the  radius,  />,  and  consequently  the  angle 
OBC,  do  not,  in  general,  coincide  with  the  vertical  plane  through  B\,  for  the 
plumb  line  does  not  point  toward  the  center  of  the  earth,  except  at  the 
poles  and  at  points  on  the  equator.  The  actual  parallax  in  zenith  distance 
is  therefore  slightly  different  from  that  given  by  (9),  and  in  addition,  there 


20 


PRACTICAL  ASTRONOMY 


is  a  minute  component  affecting-  the  azimuth.  The  influence  of  the  spheroidal 
form  of  the  earth  is  so  slight,  however,  that  it  requires  consideration  only 
in  the  most  precise  investigations. 

Finally,  it  should  be  remarked  that  the  apparent  zenith  distance  used 
for  the  calculation  of  the  parallax  is  the  observed  zenith  distance  freed  from 
refraction;  that  is,  of  the  two  corrections,  refraction  is  to  be  applied  first. 
The  zenith  distance  thus  corrected  serves  for  the  calculation  of  the  parallax. 

For  the  first  system  of  coordinates,  therefore,  and  the  limits  of  precision 
here  considered,  the  influence  of  both  refraction  and  parallax  is  confined  to 
the  coordinate  altitude,  or  its  alternative,  zenith  distance.  Hour  angle,  right 
ascension,  and  declination  are  all  affected  by  both  refraction  and  parallax, 
but,  as  these  coordinates  do  not  appear  as  observed  quantities  in  the  problems 
with  which  we  are  concerned,  the  development  of  the  expressions  which  give 
the  corresponding  corrections  is  omitted. 

TABLE  I.     MEAN  REFRACTION  AND  SOLAR  PARALLAX 
Barometer,  29.5  in.;  Thermometer,  50°  Fahr. 


h' 

z' 

r 

/ 

r-p 

h' 

z' 

r 

k   / 

r  —  p 

i5° 

75° 

3-  '5 

8'.'5 

3-  '4 

40° 

50° 

i  .'i 

6'.'7 

I'.O 

20 

70 

2.6 

8-3 

2-5 

50 

40 

0.8 

5-7 

0.7 

25 

65 

2.O 

8.0 

1.9 

60 

30 

0.6 

4-4 

o-5 

30 

60 

1.6 

7-6 

i-5 

70 

20 

0.4 

3-o 

0-3 

35 

55 

i-3 

7-2 

1.2 

80 

IO 

O.2 

i-5 

0.  I 

40 

50 

1.  1 

6.7 

I  .O 

90 

0 

0.0 

o.o 

o.o 

The  Refraction,  ;-,  and  the  Refraction — Solar  Parallax,  r-p,  are  to  be  subtracted  from  h', 
or  added  to  z'. 

The  Solar  Parallax,  /,  is  to  be  added  to  //',  or  subtracted  from  z'. 


CHAPTER    II 

FORMULA  OF  SPHERICAL  TRIGONOMETRY— TRANSFORMATION 

OF  COORDINATES— GENERAL   DISCUSSION 

OF  PROBLEMS. 

10.  The  fundamental  formulae  of  spherical  trigonometry.— Transform- 
ations of  coordinates  are  of  fundamental  importance  for  the  solution  of  most 
of  the  problems  of  spherical  and  practical  astronomy.  The  relations  between 
the  different  systems  should  therefore  receive  careful  attention.  The  more 
complicated  transformations  require  the  solution  of  a  spherical  triangle,  and, 
because  of  this  fact,  a  brief  exposition  of  the  fundamental  formulae  of  spherical 
trigonometry  is  introduced  at  this  point. 


Fig.  3- 

Let  ABC,  Fig.  3,  be  any  spherical  triangle.  Denote  its  angles  by  A,  B, 
and  C;  and  its  sides  by  a,  b,  and  c.  With  the  center  of  the  sphere,  0,  as  origin, 
construct  a  set  of  rectangular  coordinate  axes,  XYZ,  such  that  the  XY  plane 
contains  the  side  c,  and  the  ^faxis  passes  through  the  vertex  B.  Let  the  rec- 
tangular coordinates  of  the  vertex  £7  be  x,  y,  and  z.  Their  values  in  terms  of 
the  parts  of  the  triangle  and  the  radius  of  the  sphere  are 


x  =  rcosa, 

y  =  r  sin  a  cos  B, 

z  =  r  sin  a  syi  B, 


(10) 


Construct  a  second  set  of  axes,  XYZ,  with  the  origin  at  0,  the  XY  plane 
coinciding  with  the  side  c,  and  the  X  axis  passing  through  the  vertex  A.  Let 
the  coordinates  of  (Preferred  to  this  system  be  x',y,  and  z.  We  then  have 


x  =      r  cos  b, 

y  ==  —  r  sin  b  cos  A, 

J  =      r  sin  b  sin  A, 


(n) 


The  second  set  of  rectangular  axes  can  be  derived  from  the  first  by  rotat- 
ing the  first  about  the  Zaxis  through  the  angle  c.     The  coordinates  of  the  first 

21 


22  PRACTICAL  ASTRONOMY 

system  can  therefore  be  expressed  in  terms  of  those  of  the  second  by  means  of 
the  relations 

x  =  x'cos  c  — y  sin  c, 

y  =  x  sin  c  +y  cos^r,  (12) 

z  =  z'.' 

Substituting  into  equations  (12)  the  values  of  x, y,  z,  *',  /,  and  z  from  (10) 
and  (n),  and  dropping  the  common  factor  r,  we  obtain  the  desired  relations 

cos  a.  =  cos  b  cos  c  +  sin  b  sin  c  cos  A,  (13) 

sin  a  cos  B  =  cos  b  sin  c  —  sin  b  cos  c  cos  A,  (14) 

sin  as\n  B  =  sin  £sin  A.  (15) 

These  equations  express  relations  between  five  of  the  six  parts  of  the 
spherical  triangle  ABC,  and  are  independent  of  the  rectangular  coordinate 
axes  introduced  for  their  derivation.  Although  the  parts  of  the  triangle  in 
Fig.  3  are  all  less  than  90°,  the  method  of  development  and  the  results  are 
general,  and  apply  to  all  spherical  triangles.  These  relations  are  the  funda- 
mental formulae  of  spherical  trigonometry.  From  them  all  other  spherical 
trigonometry  formulas  can  be  derived.  They  determine  without  ambiguity  a 
side  and  an  adjacent  angle  of  a  spherical  triangle  in  terms  of  the  two  remain- 
ing sides  and  the  angle  included  between  them,  provided  the  algebraic  sign  of 
the  sine  of  the  required  side,  or  of  the  sine  or  cosine  of  the  required  angle, 
be  known.  Otherwise  there  will  be  two  solutions. 

Equations  (i3)-(i5)  are  conveniently  arranged  as  they  stand  if  addition- 
subtraction  logarithms  are  to  be  employed  for  their  calculation.  For  use  with 
the  ordinary  logarithmic  tables,  they  should  be  transformed  so  as  to  reduce  the 
addition  and  subtraction  terms  in  the  right  members  of  (13)  and  (14)  to  single 
terms  (Num.  Cat.  pp.  13  and  14). 

Aside  from  the  case  covered  by  equations  (i3)-(i 5),  two  others  occur  in 
connection  with  the  problems  of  practical  astronomy,  viz.,  that  in  which  the 
given  parts  are  two  sides  of  a  spherical  triangle,  and  an  angle  opposite  one  of 
them,  to  find  the  third  side;  and  that  in  which  the  three  sides  are  given,  to 
find  one  or  more  of  the  angles.  The  first  of  these  can  be  solved  for  those 
cases  which  arise  in  astronomical  practice  by  a  simple  transformation  of  (13), 
the  details  of  which  will  be  considered  in  connection  with  the  determination 
of  latitude.  A  solution  for  the  third  case  can  also  be  found  by  a  rearrange- 
ment of  the  terms  of  (13).  Thus, 

cos  a  —  cos  b  cos  c 

cos  A  =  -     —. — =— r—       — .  (16) 

sin  b  sin  c 

Similar  expressions  for  the  angles  B  and  C  can  be  derived  by  a  simple  permu- 
tation of  the  letters  in  (16).  Equation  (16)  affords  a  theoretically  accurate 
solution  of  the  problem;  but,  practically,  the  application  of  expressions  of  this 
form  is  limited  on  account  of  the  necessity  of  determining  the  angles  from 


SPHERICAL  TRIGONOMETRY  23 

their  cosines.  For  numerical  calculation  it  is  important  to  have  formulae  such 
that  the  angles  A,  B,  and  C  can  be  interpolated  from  their  tangents.  (Num. 
CaL  pp.  3  and  14).  The  desired  relations  can  be  derived  by  a  transformation 
of  (16),  (Chauvenet,  Spherical  Trigonometry,  §§  12  and  16-18),  giving 

tan,     A  =  -- 


sin  s  sin  (s-a) 

in  which  s  =  YZ  (a  +  b  -\-  c).  Similar  expressions  for  B  and  C  can  be  derived 
by  a  permutation  of  the  letters  of  (17).  When  the  three  angles  of  a  spherical 
triangle  are  to  be  determined  simultaneously,  it  is  advantageous  to  introduce 
the  auxiliary  K,  defined  by  the  relation 

sin  (s-a)  sin  (s-b)  sin  (s-c)  ,  _, 

sin  s 

Substituting  (18)  into  (17),  we  find 

K 

ian*4  A  = -. — -. r.  (19) 

sin  (s-a) 

The  expressions  tan  l/2B  and  tan  y2C are  similar  in  form. 

Collecting  results,  the  complete  formulae  for  the  calculation  of  the  three 
angles  of  a  spherical  triangle  from  the  three  sides  are 

Form  s-a,  s-b,  and  s  -  c,  and  check  by 

(s-a)  +  ( s  -  b)  -f  (s-c)  =  s. 
,.        sin  (s-a)  sin  (s-b)  sin  (s-c) 


sn  s 


,. 


K  K  K 

tan    -z  A  =  -^-.  -  -,     tan  yz  B  —  -^—,  -   ,      tan  %  C=  -r 


;.        tan    72   u   : ,          j\,         LOU    72    *--  /  \> 

sin  (s-a)  sin  (s-b)  sin  (s-c) 

/^ 
Check:     tan  %  A  tan  V2  B  tan  V2  C  =  — 


sin  5 

Two  solutions  are  possible.      The    ambiguity    is  removed  if  the  quadrant  of 
one  of  the  half-angles  of  the  triangle  is  known. 

11.  Relative  positions  of  the  reference  circles  of  the  three  coordinate 
systems. — The  transformation  of  the  coordinates  of  one  system  into  those  of 
another  requires  a  knowledge  of  the  relative  positions  of  the  reference  circles 
of  the  various  systems. 

In  the  case  of  Systems  I  and  II  the  principal  secondary  circles  coincide 
by  definition.  The  fundamental  great  circles  are  inclined  to  each  other  at  an 
angle  which  is  constant  and  equal  to  the  complement  of  the  latitude  of  the 
place  of  observation.  The  proof  of  this  statement  can  be  derived  from  Fig.  4, 
which  represents  a  section  through  the  earth  and  the  celestial  sphere  in  the 


24 


PRACTICAL  ASTRONOMY 


plane  of  the  meridian  of  the  point  of  observation,  0.  The  outer  circle  repre- 
sents the  celestial  meridian,  and  the  inner,  the  terrestrial  meridian  of  0,  the 
latter  being  greatly  exaggerated  with  respect  to  the  former.  Z  and  N  are  the 
zenith  and  the  nadir;  P  and  P',  the  poles  of  the  celestial  sphere;/  and/*',  the 
poles  of  the  earth;  HH'  and  EE,  the  lines  of  intersection  of  the  planes  of 
horizon  and  equator,  respectively,  with  the  meridian  plane,,  The  plane  of  the 
celestial  equator  coincides  with  that  of  the  terrestrial  equator,  which  cuts  the 
terrestrial  meridian  in  ee. 


But, 


whence 


Fig.  4- 
Now,  by  definition  the  arc  eO  measures   the  latitude,  y>,  of  the  point  0 


Arc  EZ=  Arc  eO 
H'E  =  90°  —  ( 


(21) 

(22} 


which  was  to  be   proved.     It  thus  appears  that   the   second  system  can  be 
derived  from  the  first  by  rotating  the  first  about    an  axis  passing  through  the 
east  and  west  points,  through  an  angle  equal  to  the  co-latitude  of  the  place. 
It  is  to  be  noted,  further,  that 


and 


Arc  ZP  —  90°  —  (p  =  Co-latitude  of  0, 
Arc  HP  =  <<>. 


(23) 
(24) 


From  (21)  and  (24)  it  follows  that  the  latitude  of  any  point  on  the  earth 
is  equal  to  the  declination  of  the  zenith  of  that  point.  It  is  also  equal  to 
the  altitude  of  the  pole  as  seen  from  the  given  point. 

Systems  II  and  III  have  the  same  fundamental  great  circle,  viz.,  the 
celestial  equator.  The  principal  secondary  of  the  third  system  does  not  main- 


RELATIVE  POSITION  OF  COORDINATE  SYSTEMS  25 

tain  a  fixed  position  with  respect  to  that  of  the  first,  but  rotates  uniformly  in  a 
clockwise  direction  as  seen  from  the  north  side  of  the  equator. 

Let  Fig.  5  represent  an  orthogonal  projection  of  the  celestial  sphere  upon 
the  plane  of  the  equator  as  seen  from  the  North.  Pis  the  north  celestial  pole; 
M,  the  point  where  the  meridian  of  0  intersects  the  celestial  equator;  and  V, 
the  vernal  equinox.  The  arc  MBV  therefore  measures  the  instantaneous 
position  of  the  principal  secondary  of  the  third  system  with  respect  to  that  of 
the  first.  This  arc  is  equal  to  the  hour  angle  of  the  vernal  equinox,  or  the 
right  ascension  of  the  observer's  meridian.  It  is  called  the  Sidereal  Time  =  6. 
We  thus  have  the  following  important  definition: 

The.  sidereal  time  at  any  instant  is  equal  to  the  hour  angle  of  the 
vernal  equinox  at  that  instant.  It  is  also  equal  to  the  right  ascension  of 
the  observer's  meridian  at  the  instant  considered. 

It  follows,  therefore,  that  the  third  system  can  be  derived  from  the  second 
by  rotating  the  second  system  about  the  axis  of  the  celestial  sphere  through 
an  angle  equal  to  the  sidereal  time. 


Finally,  the  third  system  can  be  derived  from  the  first  by  rotating  the  first 
into  the  position  of  the  second,  and  thence  into  the  position  of  the  third. 

Briefly  stated,  the  transformation  of  coordinates  involves  the  determina- 
tion of  the  changes  arising  in  the  coordinates  as  a  result  of  a  rotation  of  the 
various  systems  in  the  manner  specified  above.  It  is  at  once  evident  that  the 
transformation  of  azimuth  and  altitude  into  hour  angle  and  declination  requires 
a  knowledge  of  the  latitude;  of  hour  angle  and  declination  into  right  ascension 
and  declination,  a  knowledge  of  the  sidereal  time;  while,  to  pass  from  azimuth 
and  altitude  to  right  ascension  and  declination,  both  latitude  and  sidereal  time 
are  required.  It  is  scarcely  necessary  to  add  that  the  reverse  transformations 
demand  the  same  knowledge. 

12.  Transformation  of  azimuth  and  zenith  distance  into  hour  angle 
and  declination. — The  transformation  requires  the  solution  of  the  spherical 
triangle  ZPO,  Fig.  i,  p.  11.  The  essential  part  of  Fig.  I  is  reproduced  in  Fig.  6 
upon  an  enlarged  scale.  An  inspection  of  the  notation  of  p.  II  shows  that  the 
parts  of  the  triangle  ZPO  can  be  designated  as  shown  in  Fig.  6. 


26 


Assuming  the  latitude,  <f>,  to  be  known,  it  is  seen  that  the  transformation 
in  question  involves  the  determination  of  the  side  TT  =  90°  —  d  and  the  adja- 
cent angle  t  in  terms  of  the  other  two  sides,  96°  —  tp  and  z  =  90°  -  h,  and 
the  angle  180°  —  A  included  between  them.  Equations  (i3)-(i5)  are  directly 
applicable,  and  it  is  only  necessary  Jbo  make  the  following  assignment  of  parts: 


A  =  iSo°  — 


(25) 


c  =  90°  —  <p. 


Fig.  6. 

The  substitution  of  (25)  into  (13),  (14),  and  (15)  gives 

sin  8  =  cos  z  sin  <p  —  sin  z  cos  <p  cos  A, 
cos  d  cos  t  =  cos  z  cos  <p  -J-  sin  z  sin  <f>  cos  A, 
cos  d  sin  /  =  sin  z  sin  A. 


(26) 
(27) 

(28) 


To  adapt  these  formulae  for  use  with  the  ordinary  logarithmic  tables,  the 
auxiliary  quantities  nt  and  M,  defined  by 

m  sin  M  =  sin  z  cos  A, 
m  cosM= 


are  introduced  (Num.  Cal.  p.  14). 

Substituting  these  relations  into  (26)  and  (27)   and  collecting  results,    we 
have  for  the  calculation 


m  sin  M  =  sin  z  cos  A, 

mcosM=  cos  2, 
cos  d  sin  /   =  sin  2  sin  A, 
cos  d  cos/  —  m  cos  (<p  -  M), 
sin  d  =  m  sin  (up  -  M). 


(29) 


In  formulae  (26)-(28)  we  have  three  equations  for  the  determination  of  two 
unknown  quantities,  t  and  dj  in  (29),  five  equations  are  given  for  the  determin- 


ORDER  OF  SOLUTION  27 

ation  of  the  four  unknowns,  J/,  m,  t,  and  d.  In  both  cases  one  more  condition 
is  available  than  is  required  for  the  theoretical  solution  of  the  problem,  a  point 
of  great  practical  importance,  as  it  affords  a  means  of  testing  the  accuracy  of 
the  numerical  solution. 

The  order  of  solution  is  as  follows:  First,  determine  M  and  m  from  the 
first  two  equations.  Whatever  the  values  of  z  and  A,  there  will  always  be  two 
pairs  of  values  of  M  and  m  satisfying  these  equations.  For  one,  m  will  be  pos- 
itive; for  the  other,  negative.  It  is  immaterial,  so  far  as  the  final  values  of  / 
and  d  are  concerned,  which  of  the  two  solutions  we  adopt.  For  simplicity, 
however,  we  assume  that  m  is  always  positive.  This  makes  the  algebraic  signs 
of  sin  M  and  cosM  the  same  as  those  of  the  right-hand  members  of  the  first 
and  second  equations,  respectively,  of  (29).  The  difference  of  the  logarithms 
of  the  right-hand  members  of  these  equations  equals  log  tan  M,  from  which 
the  angle  Mis  determined,  the  quadrant  being  fixed  by  a  consideration  of  the 
algebraic  signs  of  any  two  of  the  three  functions,  sin  M,  cos  M,  and  tanM, 
M,  log  sin  M,  and  log  cos  M  are  interpolated  with  a  single  opening  of  the 
table.  The  difference  of  the  last  two  must  equal  log  tan  M,  which  affords  a 
partial,  check.  The  subtraction  of  log  sin  M  from  log  m  sin  M  gives  log  m. 
The  addition  of  this  result  to  log  cos  M  must  agree  with  the  value  of  log  m- 
cosM  from  the  second  of  (29),  which  gives  a  second  partial  check.  The  values 
of  M  and  m  thus  derived  are  to  be  substituted,  along  with  z  and  A,  into  the 
right-hand  members  of  the  last  three  of  (29)  for  the  completion  of  the  calcula- 
tion. The  left  members  of  the  third  and  fourth  of  (29)  are  of  the  same  form 
as  the  first  two,  which  makes  it  possible  to  determine  /and  cos  d  by  an  applica- 
tion of  the  process  employed  for  finding  m  and  J/,  care  being  taken  to  apply 
the  checks  at  the  points  indicated  above.  The  algebraic  sign  of  cos  3  is  nec- 
essarily positive,  since  d  must  always  lie  between  +90°  and  — 90°,  which  fixes 
the  quadrant  of  /.  It  is  to  be  noted  that  this  limitation  upon  the  sign  of  cos  3 
removes  the  ambiguity  existing  in  the  solution  of  the  general  spherical  triangle 
which  was  mentioned  on  p.  22.  The  hour  angle,  /,  and  log  cos  3  having  been 
found,  the  next  step  is  the  determination  of  log  sin  d  from  the  last  of  (29). 
The  values  of  log  cos  d  and  log  sin  3  must  correspond  to  the  same  angle. 
This  affords  a  third  partial  check.  The  determination  of  3  and  the  application 
of  the  check  can  be  accomplished  in  either  of  two  ways:  We  may  interpolate 
3  from  the  smaller  of  the  two  functions  log  cos  3  and  log  sin  3,  and  check  by 
comparing  the  other  function  with  the  value  interpolated  from  the  tables  with 
the  calculated  3  as  argument;  or  we  may  interpolate  3  from  log  tan  3,  which  is 
found  by  subtracting  log  cos  3  from  log  sin  3.  With  the  value  of  3  thus  de- 
rived, log  sin  3  and  log  cos  3  are  interpolated  from  the  table.  The  interpolated 
values  must  agree  with  those  resulting  from  the  last  three  equations  of  (29). 
The  former  method  is  shorter;  the  latter,  more  precise  in  the  long  run,  although 
not  necessarily  so  in  any  specific  case.  In  practice,  the  first  method  is  usually 
sufficient. 

In  applying  the  checks  it  is  to  be  noted  that  the  accumulated  error  of  cal- 
culation (Num.  Cal.  pp.  4  and  12)  may  produce  a  disagreement  of  one,  and  in 
rare  instances,  of  two  units  in  the  last  place  of  decimals.  Great  care  must  be 


PRACTICAL  ASTRONOMY 


exercised  with  the  algebraic  signs  of  the  trigonometric  functions  and  in  assign- 
ing the  quadrants  of  the  angles.  Otherwise,  an  erroneous  computation  may 
apparently  check.  The  check  quantities  must  agree  both  in  absolute  magni- 
tude and  algebraic  sign. 

The  calculation  of  t  and  3  from  equations  (26)-(28)  with  the  aid  of  addition- 
subtraction  logarithms  is  accomplished  by  an  application  of  the  method  used 
for  the  solution  of  the  last  three  of  (29).  The  only  differences  which  occur  are 
to  be  found  in  the  details  of  the  combination  of  the  quantities  which  enter  into 
the  right  members  of  the  two  groups  of  equations. 

Example  1.  For  a  place  of  observation  whose  latitude  is  38°  56'  51",  the  azimuth  of  an 
object  is  97°  14'  12"  and  its  zenith  distance  62°  37'  49".  Find  the  corresponding  hour  angle 
and  declination. 

The  calculation  for  equations  (26)-(28),  using  addition-subtraction  logarithms,  appears 
in  the  first  column;  that  for  equations  (29),  made  with  the  ordinary  tables,  is  in  the  second 
column.  For  the  first,  3  is  derived  from  log  sin  #,  which,  in  this  case,  is  smaller  than  log  cos  $. 
In  the  calculation  of  (29),  d  is  determined  from  log  tan  5.  The  arguments  tor  the  check 
quantities,  sin  5  and  cos<J,  need  not  ordinarily  be  written  down.  They  are  inserted  here  In 
order  to  illustrate  the  application  of  the  control.  The  abbreviation  log  is  not  prefixed  to  the 
arguments,  although  the  majority  of  the  numbers  appearing  in  the  computation  are  logarithms. 
Its  omission  saves  time  and  produces  no  confusion. 


97°  H'  12" 
62   37  49 
38  56  5' 


sin  A 
sin  z 
cos  A 
sin  <p 
cos  y 
cosz 

cosz  sin  V 

sin  z  cos  <p  cos  ^4 

A 

B 

cos  z  co<f 

sin  z  sin  <p  cos  A 

B 

A 

cos  (Jsin  / 

cos  d  cos  / 

tan/ 


sin/ 
cos  / 
cos  d 
sin  5 
5  (from  sin  <*) 


9.94844 

9.10026* 

9.79838 

9.89082 

9.66250 

9.46088 

8.93952« 

9.47864 

0.11430 

9-5S332 

8.84708,, 

0.70624 

o. 61112 

9-94497 

9.45820 

0.48677 

4»47'»46!4 
9 . 97806 
9.49130 
9.96691 
9-575I8 

h22°S'S" 

9.96660  Ck. 


sin  A 

sin  z 

cos  A 

m  sin  M 

m  cos  M 

tan  M 

M 

¥ 

<P—M 
sin  M 
co&M 
log  m 

sin  (^  —  Af~) 

cos(?-AO 

cos  ^sin  / 

cos  ^  cos / 

tan/ 

/ 

/ 

sin  / 
cos  / 
cos  5 
sin  d 
tan^ 
d 

sin  8 
cos  5 


9-94844 

9. IOO26« 

9  04870* 
9 . 66250 
9.38620,, 
-  I3°40'34" 
385651 
52  37  25 
9-37371* 
9.98751 
9.67499 
9.90018 
9.78322 

9-94497 
9.45821 

0.48676 
7i°56'35" 
4h47m46f3 
9 . 97806 

9-49I3I 
9.96691 
9-575I7 
9.60826 

+22°5'5" 

9.57517  Ck. 
9.96691  Ck. 


29 

13.  Transformation  of  hour  angle  and  declination  into  azimuth  and 
zenith  distance,  —  The  transformation  can  be  effected  by  solving  (29)  in  the 
reverse  order  to  that  followed  in  Section  12.  It  is  better,  however,  to  use  equa- 
tions of  the  same  form  as  those  appearing  in  the  preceding  section,  thus  re- 
ducing the  two  problems  to  the  same  type.  As  before,  two  sides  and  the  in- 
cluded angle  are  given,  to  find  the  remaining  side.  With  the  following  assign- 
ment of  parts 

a  •=•  5,  A  =  /, 

£  =  go°  —  «,          5=i8o°  —  A,  (30) 

c  —  90°  —  <p, 

we  find  by  substituting  into  (13),  (14),  and  (15), 

cos.3—       sin  d  s'\n  <p  -f-  coso  cos  ^  cos/,  (31) 

sin  z  cos  A  —  —  sin  d  cos  y  +  cos  d  sin  <p  cos  /,  (32) 

coso  sin/.  (33) 


These  are  of  the  same  general  form  as  (26),  (27),  and  (28).  Applying  the 
same  principle  as  before,  we  derive 

n  sin  N  =  sin  d, 

n  cos  N  =  cos  3  cos  /, 

sin  z  sin  A  =  cos  d  sin  /,  (34) 

sin  z  cos  A  =  nsin  (<p  -  N), 
cos  z  =  n  cos  (<p  -  N}. 

The  two  groups  (3i)-(33)  and  (34)  give  the  required  transformation.  The 
former  can  be  used  with  addition-subtraction  logarithms;  the  latter,  with  the 
ordinary  tables.  A  comparison  of  these  equations  with  groups  (26)-(28)  and 
(29)  shows  that  the  same  arrangement  of  calculation  can  be  used  for  both 
transformations.  The  unknowns  are  involved  in  the  same  manner  in  both 
cases,  with  the  exception  that  the  sine  and  cosine  of  z  are  interchanged  in  the 
left  members  of  (3i)-(34)  as  compared  with  the  corresponding  functions  of  d  in 

(26)-(29). 

In  the  solution  of  (3i)-(33)  and  (34),  the  quadrant  of  A  is  fixed  by  the  fact 
that  sin  z  is  necessarily  positive,  since  z  is  always  included  between  o°  and 
+  1  80°.  This  eliminates  the  ambiguity  attached  to  the  solution  of  the  general 
spherical  triangle. 

14.  Transformation  of  hour  angle  into  right  ascension,  and  vice 
versa.  —  Since  the  coordinate  declination  is  common  to  Systems  II  and  III, 
the  transformation  of  the  coordinates  of  one  of  these  systems  into  those  of 
the  other  requires  only  a  knowledge  of  the  relation  between  hour  angle  and 
right  ascension. 

In  Fig.  5,  p.  25,  let  B  be  the  intersection  of  the  hour  circle  through  any 
celestial  body  with  the  celestial  equator.  We  then  have  by  definition 


30 


whence 


PRACTICAL  ASTRONOMY 

Arc  MB  —  t  =  Hour  angle  of  object, 
Arc  VM  =  a=  Right  ascension  of  object, 
Arc  MVB  =  6=  Sidereal  time, 


a  ^=  9  —  /, 
t  —  9  —  a. 


(35) 
(36) 


Equations  (35)  and  (36)  express  the  required  transformations.  The  same 
result  can  be  derived  from  Fig.  i,  p.  11,  the  point  /,  in  this  figure,  correspond- 
ing to  B  in  Fig.  5. 

Example  2.  In  a  place  of  observation  whose  latitude  is  38°  58'  53",  the  hour  angle  of  an 
object  is  2Ohi9m4i»8,  and  its  declination  —  8°  31'  47".  Find  the  corresponding  azimuth  and 
zenith  distance. 

The  calculation  by  equations  (3i)-(33)  is  in  the  first  column;  that  by  equations  (34),  in 
the  second. 

/  =  2oP  i9m  4i«8  =  304°  55'  27" 
8=—  8°  31'  47" 

V=    38  58   53 


sin  / 

9-9i377» 

sin^ 

9-9i377« 

cos  8 

9  995'7 

cos  8 

9-995I7 

cos  / 

9-75777 

cos  t 

9-75777 

sin  <f 

9.79870 

n  sin  N 

9-  17121* 

COS<f> 

9.89061 

n  cos  N 

9-75294 

sin  8 

9-i7i2i« 

tan  N 

9.  41827* 

sin  <?sin  <p 

8.96991* 

N 

-  14°40'  50" 

cos  8  cos  <p  cos  t 

9-64355 

<f> 

38  58  53 

B 

0.67364 

<p-W 

53  39  43 

A 

0.57016 

sin  N 

9.40386* 

sin  8  cos  <f> 

9  06182* 

cos  N 

9-98559 

cos  8  sin  <f>  cos  t 

9-55I64 

log» 

9-76735 

A 

9.51018 

sin  (<p-  N) 

9.90608 

B 

0.12180 

cos  (<p  -  N) 

9-77273 

sin  z  sin  A 

9.90894* 

sin  z  sin  A 

9.90894* 

sin  z  cos  A 

9-67344 

sin  z  cos  A 

9-67343 

tan  A 

o-2355°« 

tan  A 

0.23551* 

A 

300°  10'  3  1" 

A 

300°  10'  29" 

sin  A 

9.93676* 

sin  A 

9-93677« 

cos  A 

9.70126 

cos  A 

9.70126 

sin  z 

9.97218 

sin  z 

9.97217 

cosz 

9.54007 

COS  2 

9.54008 

z  (from  cos*) 

69°  42  '32" 

z  (from  cos  z) 

69°  42  '30" 

sin  z 

9.97218  Ck. 

sin  z 

9.97217  Ck 

TRANSFORMATIONS  31 

Example  3.     What  is  the  right  ascension  of  an  object  whose  hour  angle  is  I7h2im34«6, 
when  the  sidereal  time  is  2ihi4m52«8? 

By  equation  (35) 

0  =  2lhI4m52«8 

*  =  17  21   34-6 

a=    3  53    18.2,  Ans. 

Example  4.    What  is  the  hour  angle  of  an  object  whose  right  ascension  is  8hi2m34*8, 
when  the  sidereal  time  is  &6m28*7? 

By  equation  (36) 


a  =    8  12   34.8 

t  =  21  53   53-9>  Ans- 

15.  Transformation  of  azimuth  and  altitude  into  right  ascension  and 
declination,  or  vice  versa.  —  These  transformations  are  effected  by  a  combina- 
tion of  the  results  of  Sections  12-14.     For  the  direct  transformation,  deter- 
mine /  and  S  by  (26)-(28)  or  (29),  and   then   a    by  (35).      For   the   reverse 
calculate  t  by  (36),  and  then  A  and  z  by  (3i)-(33)  or  (34). 

Example  5.  What  is  the  right  ascension  of  the  object  whose  coordinates,  at  the  sidereal 
time  I7h2imi6'4,  are  those  given  in  Example  i? 

The  hour  angle  found  in  the  solution  of  Example  i  by  equations  (26)-(28)  is  4h47m46»4. 
This,  combined  with  0  =  I7h2imi6»4  in  accordance  with  equation  (35^,  gives  for  the  required 
right  ascension  I2h33m3o;o. 

Example  6.  At  a  place  whose  latitude  is  38°  38'  53",  what  are  the  azimuth  and  zenith  dis- 
tances of  an  object  whose  right  ascension  and  declination  are  9h27mi4«2  and  —  8°  31  '47",  re- 
spectively, the  sidereal  time  being  5h46m56fo? 

By  equation  (36;,  /  =  2ohi9m4if8.    We  have,  further,  d  =  —  8°  31  '47"  and  <f>  =  38°  38'  53". 
These  quantities  are  the  same  as  those  appearing  in  Example  2.     The   solution    by  equa- 
tions (39)  gave  A  —  300°  10'  29",  z  =  69°  42'  30". 

16.  Given  the  latitude  of  the  place,  and  the  declination  and  zenith 
distance  of  an  object,  to  find  its  hour  angle,  azimuth,  and  parallactic 
angle.  —  We  have  given  three  sides  of  the  spherical  triangle  ZPO,  Fig.  6,  p.  26* 
to  find  the  three  angles,  the  parallactic  angle  being  the  angle  at  the  object. 
The  parallactic  angle  is  not  used  in  engineering  astronomy,  although  its  value 
is  frequently  required  in  practical  astronomy  proper. 

Equations  (20)  are  directly  applicable  for  the  solution  of  the  problem. 
Assigning  the  parts  of  the  triangle  as  in  (30),  and,  further,  writing  the  angle 
C  =  q  =  parallactic  angle,  we  have  for  the  calculation. 


32  PRACTICAL  ASTRONOMY 

a  =  £,     b  =  90°  —  o,     c  =  go0  — 

s  =  y-z  (a  +  b  -f  4 
Check:     (j  -a)  +  (.r-£)  +  ($-*•)  =  j, 


A-*-M 

ii  ^  —  a)  sin  \s-vj  sin  ^j—  cj 

sin  s 

K                                     fC 

sin  (s-a)                         sin  (s-t>) 

A" 


sin 


Check:     tan  ^  /  cot  y2  A  tan  ^  q  =  — 


sin 
Object  ^    wcsl  }•  of  meridian,  Yz  t,  %A,  %  q  in 


In  engineering  astronomy  the  determination  of  the  hour  angle,  A  is  usually 
all  that  is  required.  For  this  case  it  is  simpler  to  use  equation  (17).  The 
formulae  are 

a  =  z,      b  =  90°  -  o,     c  =  90°  -  <f>, 

s=y2(a  +  b  +  c). 
Check:   (s  -  a)  +  (s  -  b}  +  (s  -  c)  =  s,  (38) 

3n,  v  f  _  sin  (s  -  b}  sin  (s  -  c) 

tan   YZ  t  — -. ; — . r — 

sin  s  sin  (s  -  a) 

where  y2  t  is  to  be  taken  in  the  first  or  second  quadrant  according  as  the  ob- 
ject is  west  or  east  of  the  meridian  at  the  time  of  observation. 

For  those  cases  in  which  the  object  is  more  than  two  and  one-half  or  three 
hours  from  the  meridian,  equation  (16)  written  in  the  form 

cos  z  -  sin  dsin  <p 

cos  /  =  -        — ^—          -s  (39) 

cos  o  cos  <f> 

will  usually  give  satisfactory  results.  In  any  case,  (39)  affords  a  valuable  con- 
trol upon  the  value  of  /  given  by  (38).  The  numerator  of  the  right  member  of 
(39)  is  readily  calculated  by  means  of  addition-subtraction  logarithms. 

17.  Application  of  transformation  formulae  to  the  determination  of 
latitude,  azimuth,  and  time. — It  was  shown  in  Section  4  that  the  solution  of 
the  fundamental  problems  of  practical  astronomy  requires  the  determination 
of  the  position  of  the  axis  of  the  celestial  sphere  and  the  orientation  of  the 
sphere  as  affected  by  the  diurnal  rotation.  In  practice  this  is  accomplished 
indirectly  by  observing  the  positions  of  various  celestial  bodies  with  respect  to 
the  horizon,  the  observed  data  being  combined  with  the  known  position  of  the 
bodies  on  the  sphere  for  the  determination  of  the  position  of  the  sphere  itself. 
The  means  for  effecting  the  coordinate  transformation  hereby  implied  are  to 
be  found  in  the  formulae  of  Sections  12-16. 

Although  the  most  advantageous  determination  of  latitude,  azimuth,  and 
time  requires  a  modification  of  these  formulae,  it  is,  nevertheless,  easy  to  see 
that  the  solution  of  the  various  problems  is  within  our  grasp,  and  that  the 


TRA  NSFORMA  TIONS—  GENERA  L  DISC  US  SI  ON 


33 


Example  7.  For  a  place  whose  latitude  is  38°56'5i",  find  the  hour  angle,  azimuth,  and 
parallactic  angle  of  an  object  east  of  the  meridian  whose  declination  and  zenith  distance  are 
—  8°  16'  14"  and  54°  16'  12",  respectively. 

Equations  (37)  are  used  for  the  solution,  which  is  given  below  in  the  column  on  the  left. 
If  only  the  hour  angle  were  required,  equations  (38)  or  (39)  would  be  used.  As  an  illustra- 
tion of  the'  application  of  these  formulae,  the  problem  is  also  solved  on  this  assumption.  The 
first  ten  lines  of  the  computation  for  (38),  being  the  same  as  that  for  (37),  are  omitted.  The 
remainder  of  the  calculation  for  (38)  occupies  the  upper  part  of  the  right-hand  column.  The 
solution  by  (39)  is  in  the  lower  part  of  this  column.  The  object  is  rather  too  near  the  merid- 
ian for  the  satisfactory  use  of  equation  (39),  although  it  happens  that  the  resulting  value  of 
the  hour  angle  agrees  well  with  that  from  (37)  and  (38). 


tan 


8 


b 
c 

25 
S 

s  -  a 

s-b 

s  -  c 

sin  (s  -  «) 

sin  (5  -  b) 

sin  (5  -  c) 

cosec  5 


tan  yz  t 

cot  }£  A 

tan  %  q 

i  co\.yzA  t&n%g 
K  cosec 5 

y*t 


—8°  16'  14" 

38  56  51 

54  16  12 

98  16  14 

5i  3  9 

203  35  35 

101  47  48  Ck. 

47     3i     36 

3    3i     34 
50    44    39 

9.86782 

8.78890 

9.88892 

0.00927 

8-55491 


sin  (.?-£) 

sin  (s-c) 

cosec  (s-a~) 

cosec  s 

tan*  %  t 

tan      t 


8.78890 
9.88892 
0.13218 
0.00927 
8.81927 
9.40964,, 

165°  35'   46' 
33i     ii     32 


9.40964,, 
0.48856* 

9-38854* 
9.28674* 
9.28673,, 


Ck. 


165' 

162 

1 66 

33i 

324 

332 


35 
o 

15 

i 
30 


46" 


36 

20 


sin  d 
sin  y 
cos  8 

COS  (ft 

cos  z 

sin  d  sin  ,_- 
A 
B 

cos  z  —  sin  d  sin 
cos  d  cos  <p 
cos  t 
t 


9-I5790* 
9.79838 
9.99546 
9.89082 
9.76639 
8.95628,, 
9.18989 
0.06252 
9.82891 
9.88628 
9.94263 
331°  ii'  34' 


adaptation  of  the  equations  to  any  special  case  is  only  a  matter  of  detail. 
Consider  for  a  moment  either  equations  (26)-(28)  or  (3i)-(33).  Both  groups 
involve  the  five  quantities  A,  z,  t,  d,  and  <p;  but,  since  /  =  d  -  a,  we  may  regard 
them  as  functions  of  the  six  quantities  A,  z,  a,  d,  <p,  and  6.  If,  therefore,  the 
zenith  distance  of  a  star  of  known  right  ascension  and  declination  be  measured, 
either  group  will  enable  us,  theoretically  at  least,  to  determine  <p,  A,  and  6 — 
the  latitude,  the  azimuth  of  the  star,  and  the  sidereal  time.  The  azimuth  of 
the  star  being  known,  the  azimuth  of  any  other  object,  a  distant  terrestrial 
mark,  for  example,  can  be  found  by  applying  to  the  calculated  position  of  the 
star  the  difference  in  azimuth  of  the  star  and  the  mark.  The  latter  can  be  ob- 
served directly  with  any  instrument  adapted  for  the  measurement  of  horizontal 
angles.  Further,  the  sidereal  time,  as  will  be  shown  in  Chapter  III,  bears  an 
3 


34  PRACTICAL  ASTRONOMY 

intimate  relation  to  all  of  the  other  kinds  of  time,  so  that,  if  the  sidereal  time 
has  been  found,  the  determination  of  the  others  becomes  but  a  matter  of  cal- 
culation. 

Practically,  such  a  solution  would  be  complicated.  It  is  simpler  to  deter- 
mine <p,  A,  and  Q  separately,  assuming  for  the  calculation  of  each  that  one  or 
both  of  the  others  are  known. 

For  example,  equation  (31)  is  a  function  of  #,  <5,  ^,  and  t  —  6  -  a.  Let  it 
be  assumed  that  the  zenith  distance  of  a  star  of  known  right  ascension  and 
declination  has  been  measured  and  that  the  time  of  observation  has  been 
noted.  The  substitution  of  the  resulting  data  into  (31)  leads  to  the  determin- 
ation of  the  only  remaining  unknown,  namely,  the  latitude,  <f>. 

Again,  the  elimination  of  z  from  (32)  and  (33)  gives  an  expression  for  A 
as  a  function  for  <f>,  d,  and  t  =  6  -  a.  Let  it  be  assumed  that  tp  and  6  are  known. 
The  azimuth  of  a  star  of  known  right  ascension  and  declination  can  therefore 
be  calculated.  The  calculated  azimuth  applied  to  the  observed  difference  in 
azimuth  of  star  and  mark  gives  the  azimuth  of  the  mark. 

Finally,  equations  (38)  and  (39)  express  the  hour  angle,  t,  as  a  function  of 
#,  ^,  and  d.  If  the  zenith  distance  of  a  star  of  known  right  ascension  and  dec- 
lination be  measured  in  a  place  of  known  latitude,  the  hour  angle  can  be  cal- 
culated. Equation  (35),  in  the  form  6  =  t  +  a,  then  gives  the  sidereal  time  of 
observation. 

The  solutions  thus  outlined  require,  for  the  determination  of  latitude,  a 
knowledge  of  the  time;  for  the  determination  of  time,  a  knowledge  of  the  lat- 
itude; and,  for  azimuth,  both  time  and  latitude.  For  the  first  two,  time  and 
latitude,  it  might  appear  that  the  methods  proposed  are  fallacious.  If  each  is 
required  for  the  determination  of  the  other,  how  can  either  ever  be  determined? 
The  explanation  is  to  be  found  in  the  fact  that  the  formulae  can  be  arranged 
in  such  a  way  that  an  approximate  value  for  either  of  these  quantities  suffices 
for  the  determination  of  a  relatively  precise  value  of  the  other.  Thus,  a  mere 
guess  as  to  the  time  will  lead  to  a  relatively  accurate  value  of  the  latitude, 
which,  in  turn,  can  be  used  for  the  determination  of  a  more  precise  value  of  the 
time.  The  process  can  be  repeated  as  many  times  as  may  be  necessary  to  se- 
cure the  desired  degree  of  precision.  The  principle  involved  in  the  procedure 
thus  outlined  is  called  the  Method  of  Successive  Approximations.  In 
numerical  investigations  it  is  of  great  importance.  The  method  amounts, 
practically,  to  replacing  a  single  complex  process  by  a  series,  consisting  of 
repetitions  of  some  relatively  simple  operation.  Ordinarily,  the  success  of  the 
method  depends  upon  the  number  of  repetitions  or  approximations  which 
must  be  made  in  order  to  arrive  at  the  desired  result.  If  the  convergence  is 
rapid,  so  that  one  or  two  approximations  suffice,  the  saving  in  time  and  labor 
as  compared  with  the  direct  solution  is  frequently  very  great.  Indeed,  in  some 
instances,  the  method  of  successive  approximations  is  the  only  method  of  pro- 
cedure, the  direct  solution  being  impossible  as  a  result  of  the  complexity  of 
the  relation  connecting  the  various  quantities  involved. 


GENERAL  DISCUSSION  35 

The  general  method  of  procedure  for  the  solution  of  the  problems  of 
latitude,  time,  and  azimuth  has  been  outlined.  There  remains  the  formulation 
of  the  details.  But,  before  proceeding  to  a  detailed  development,  we  must 
consider  the  subject  of  time  in  its  theoretical  aspects — the  different  kinds  of 
time,  their  definition  and  their  relations.  Chapter  III  will  be  devoted  to  this 
question.  We  must  also  consider  the  various  astronomical  instruments  that 
find  application  in  engineering  astronomy — their  characteristics  and  the  con- 
ditions under  which  they  are  employed,  since  the  nature  of  the  data  obtained 
through  their  use  will  influence  the  arrangement  of  the  solutions.  Chapter  IV 
is  therefore  devoted  to  a  discussion  of  various  astronomical  instruments. 

In  arranging  the  details  of  the  methods  for  the  determination  of  latitude, 
time,  and  azimuth,  it  is  to  be  remembered  that  the  various  problems  are  not 
merely  to  be  solved,  but  they  are  to  be  solved  with  a  definite  degree  of 
precision,  and  with  a  minimum  expenditure  of  labor.  This  requirement 
renders  the  question  one  of  some  complexity,  for  the  precision  required  may 
vary  within  wide  limits.  For  many  purposes  approximate  results  will  suffice, 
and  it  is  then  desirable  to  sacrifice  accuracy  and  thus  reduce  the  labor 
involved.  On  the  other  hand,  in  astronomical  work  of  the  highest  precision, 
no  means  should  be  overlooked  which  can  in  any  way  contribute  toward  an 
elimination  or  reduction  of  the  errors  of  observation  and  calculation. 

The  problems  with  which  we  have  to  deal  therefore  present  themselves 
under  the  most  diverse  conditions,  and,  if  an  intelligent  arrangement  of  the 
methods  is  to  be  accomplished,  one  must  constantly  bear  in  mind  the  results 
which  will  be  established  in  the  two  following  chapters,  as  well  as  those 
already  obtained  in  the  discussion  of  the  principles  of  numerical  calculation. 


CHAPTER  III 

TIME  AND  TIME  TRANSFORMATION 

18.  The  basis  of  time  measurement— The  rotation  of  the  earth  is  the 
basis  for  the  measurement  of  time.    Svmce  motion  is  relative,  we  must  specify  the 
object  to  which  the  rotation  is  referred.     By  referring  to  different  objects,  it  is 
obvious  that  we  may  have  several  different  kinds  of  time.    Actually,  the  rotation 
of  the  earth  is  referred  to  three  different  things:  the  apparent,  or  true,  sun,  a 
fictitious  object  called  the  mean  sun,  and  the  vernal  equinox.     In  practice,  how-' 
ever,  we  turn  the  matter  about  and  take  the  apparent  diurnal  rotations  of  these 
objects  with  reference  to  the  meridian  of  the  observer,  considered  to  be  fixed, 
as  the  basis  of  time  measurement.     We  have,  accordingly,  three  kinds  of  time: 
Apparent,  or  True,  Solar  Time,  Mean  Solar  Time,  and  Sidereal  Time. 

19.  Apparent,  or  True,  Solar  Time=A.S.T.— The  apparent,  or  true,  solar 
time  at  any  instant  is  equal  to  the  hour  angle  of  the  apparent,  or  true,  sun 
at  that  instant. 

The  interval  between  two  successive  transits  of  the  apparent,  or  true,  sun 
across  the  same  meridian  is  called  an  Apparent,  or  True,  Solar  Day=A.  S.  D. 

The  instant  of  transit  of  the  apparent  sun  is  called  Apparent  Noon— A.  N. 

In  astronomical  practice  the  apparent  solar  day  begins  at  apparent  noon. 
It  is  subdivided  into  24  hours,  which  are  counted  continuously  from  o  to  24. 
The  earth  revolves  about  the  sun  in  an  elliptical  orbit,  the  sun  itself  occupying 
one  of  the  foci  of  the  ellipse.  The  earth's  motion  is  such  that  the  radius  vector 
connecting  it  with  the  sun  sweeps  over  equal  areas  in  equal  times.  Since  the 
distance  of  the  earth  from  the  sun  varies,  it  follows  that  the  angular  velocity 
of  the  earth  in  its  orbit  is  variable.  Hence,  the  angular  motion  of  the  sun 
along  the  ecliptic,  which  is  but  a  reflection  of  the  earth's  orbital  motion,  is  also 
variable.  The  projection  into  the  equator  of  the  motion  along  the  ecliptic  is  like- 
wise variable,  not  only  because  the  ecliptical  motion  is  variable,  but  also  on 
account  of  the  fact  that  the  angle  of  projection  changes,  being  o  degrees  at  the 
solstices,  and  about  23^  degrees  at  the  equinoxes.  Apparent  solar  time  is 
not,  therefore,  a  uniformly  varying  quantity,  nor  are  apparent  solar  days  of 
the  same  length. 

The  adoption  of  such  a  time  system  for  the  regulation  of  the  affairs  of 
everyday  life  would  bring  with  it  many  inconveniences,  the  first  of  which  would 
be  the  impossibility  of  constructing  a  timepiece  capable  of  following  accurately 
the  irregular  variations  of  apparent  solar  time.  On  this  account  there  has  been 
devised  a  uniformly  varying  time,  based  upon  the  motion  of  a  fictitious  body 
called  the  mean  sun. 

20.  Mean  Solar  Time=M.  S.  T. — The  mean  sun  is  an  imaginary  body 
supposed  to  move  with  a  constant  angular  velocity  eastward  along  the  equator, 
such   that  it  completes  a  circuit  of  the  sphere  in  the  same  time  as  the  apparent, 
or  true,  sun.    Further,  the  mean  sun  is  so  chosen  that  its  right  ascension  differs 
as  little  as  possible,  on  the  average,  from  that  of  the  true  sun. 

36 


DEFINITIONS  37 

The  Mean  Solar  Time  at  any  instant  is  equal  to  the  hour  angle  of  the 
mean  sun  at  that  instant. 

The  interval  between  two  successive  transits  of  the  mean  sun  across  the 
same  meridian  is  called  a  Mean  Solar  Day— M.  S.  D. 

The  instant  of  transit  of  the  mean  sun  is  called  Mean  Noon=M.  N. 

Mean  solar  time  is  a  uniformly  varying  quantity  and  all  mean  solar  days  are 
of  the  same  length.  Mean  solar  time  is  the  time  indicated  by  watches  and  clocks, 
generally,  throughout  the  civilized  world,  and  the  mean  solar  day  is  the  standard 
unit  for  the  measurement  of  time. 

In  astronomical  practice  the  mean  solar  day  begins  at  mean  noon.  It  is 
subdivided  into  24  hours  which  are  numbered  continuously  from  o  to  24.  The 
astronomical  date  therefore  changes  at  noon.  But  since  a  change  of  date  during 
the  daylight  hours  would  be  inconvenient  and  confusing  for  the  affairs  of  every- 
day life,  the  Calendar  Date,  or  Civil  Date,  is  supposed  to  change  1 2  hours  before 
the  transit  of  the  mean  sun,  i.e.  at  the  midnight  preceding  the  astronomical 
change  of  date.  Further,  in  most  countries,  the  hours  of  the  civil  mean  solar 
day  are  not  numbered  continuously  from  o  to  24,  but  from  o  to  12,  ana  then 
again  from  o  to  12,  the  letters  A.  M.  or  P.  M.  being  affixed  to  the  time  in  order 
to  avoid  ambiguity.  For  example  the  civil  date  1907,  Oct.  8,  ioh  A.  M.,  is 
equivalent  to  the  astronomical  date,  1907,  Oct.  7,  22h.  The  astronomical  day 
Oct.  8  did  not  begin  until  the  mean  sun  was  on  the  meridian  on  Oct.  8  of  the 
calendar. 

From  the  manner  of  definition,  it  is  evident  that  at  any  instant  the  mean 
solar  time  for  different  places  not  on  the  same  meridian  is  different.  If  each 
place  were  to  attempt  to  regulate  its  affairs  in  accordance  with  its  own  local 
mean  solar  time,  confusion  would  arise,  especially  in  connection  with  railway 
traffic.  To  avoid  this  difficulty  all  points  within  certain  limits  of  longitude  use 
the  time  of  the  same  meridian.  The  meridians  selected  for  this  purpose  are 
all  an  exact  multiple  of  15  degrees  from  the  meridian  of  Greenwich,  with  the 
result  that  all  timepieces  referred  to  them  indicate  at  any  instant  the  same  number 
of  minutes  and  seconds,  and  differ  among  themselves,  and  from  the  local  mean 
solar  time  of  the  meridian  of  Greenwich,  by  an  exact  number  of  hours.  The 
system  thus  defined  is  called  Standard  Time. 

Although,  theoretically,  all  points  within  7^2  degrees  of  longitude  of 
a  standard  meridian  use  the  local  mean  solar  time  of  that  meridian,  actually,  the 
boundaries  separating  adjacent  regions  whose  standard  times  differ  by  one 
hour  are  quite  irregular. 

The  standard  meridians  for  the  United  States  are  75,  90,  105,  and  120 
degrees  west  of  Greenwich.  The  corresponding  standard  times  are  Eastern, 
Central,  Mountain,  and  Pacific.  These  are  slow  as  compared  with  Greenwich 
mean  solar  time  by  5,  6,  7,  and  8  hours,  respectively. 

21.  Sidereal  Time. — The  sidereal  time  at  any  instant  is  equal  to  the  hour 
angle  of  the  true  vernal  equinox  at  that  instant.  (See  p.  25.) 

The  interval  between  two  successive  transits  of  the  true  vernal  equinox 
across  the  same  meridian  is  called  a  Sidereal  Day— S.  D. 


38  PRACTICAL  ASTRONOMY 

The  instant  of  transit  of  the  true  vernal  equinox  is  called  Sidereal 
Noon^S.  N. 

Since  the  precessional  and  nutational  motions  of  the  true  equinox  are  not 
uniform,  sidereal  time  is  not,  strictly  speaking,  a  uniformly  varying  quantity, 
but  practically  it  may  be  considered  .as  such,  for  the  variations  in  the  motion 
of  the  equinox  take  place  so  slowly  that,  for  the  purposes  of  observational  astron- 
omy, all  sidereal  days  are  of  the  same  length. 

The  importance  of  sidereal  time  in  the  transformation  of  the  coordinates 
of  the  second  system  into  those  of  the  third,  and  vice  versa,  has  already  been 
shown  in  Sections  n  and  14.  It  also  plays  an  important  role  in  the  determina- 
tion of  time  generally,  for  sidereal  time  is  more  easily  determined  than  either 
apparent  or  mean  solar  time. 

The  usual  order  of  procedure  in  time  determination  is  as  follows :  Every 
observatory  possesses  at  least  one  sidereal  timepiece  whose  error  is  determined 
by  observations  on  stars.  The  true  sidereal  time  thus  obtained  is  transformed 
into  mean  solar  time  by  calculation,  and  used  for  the  correction  of  the  mean 
solar  timepieces  of  the  observatory.  Certain  observatories,  in  particular  the 
United  States  Naval  Observatory  at  Washington,  and  the  Lick  Observatory  at 
Mt.  Hamilton  in  California,  send  out  daily  over  the  wires  of  the  various  tele- 
graph companies,  series  of  time  signals  which  indicate  accurately  the  instant 
of  mean  noon.  These  signals  reach  every  part  of  the  country,  and  serve  for  the 
regulation  of  watches  and  clocks  generally. 

22.  The  Tropical  Year. — Several  different  kinds  of  years  are  employed 
in    astronomy.     The   most  important  are   the  tropical  and   the  Julian.     The 
Tropical  Year  is  the  interval  between  two  successive  passages  of  the  mean 
sun  through  the  mean  vernal  equinox.     Its  length  is  365.2422  M.  S.  D.     During 
this  interval   the  mean  sun   makes  one  circuit  of  the  celestial  sphere   from 
equinox  to  equinox  again,  in  a  direction  opposite  to  that  of  the  rotation  of  the 
sphere  itself,  whence  it  follows  that  during  a  tropical  year  the  equinox  must 
complete  366.2422  revolutions  with  respect  to  the  observer's  meridian.     We 
therefore  have  the  important  relation: 

One  Tropical  ¥63^=365.2422  M.  S.  0.^366.2422  S.  D.      (40) 

In  accordance  with  a  suggestion  due  to  Bessel,  the  tropical  year  begins 
at  the  instant  when  the  mean  right  ascension  of  the  mean  sun  plus  the  constant 
part  of  the  annual  aberration  is  equal  to  280°  or  i8h  4Om.  The  symbol  for  this 
instant  is  formed  by  affixing  a  decimal  point  and  a  zero  to  the  corresponding 
year  number;  thus  for  1909,  the  beginning  of  the  tropical  year  is  indicated  by 
1909.0.  This  epoch  is  independent  of  the  position  of  the  observer  on  the  earth 
and  does  not,  in  general,  coincide  with  the  beginning  of  the  calendar  year, 
although  the  difference  between  the  two  never  exceeds  a  fractional  part  of  a  day. 

23.  The  Calendar. — For  chronological  purposes  the  use  of  a  year  involving 
fractional  parts  of  a  day  would  be  inconvenient.     That  actually  used  has   its 
origin  in  a  decree  promulgated  by  Julius  Caesar  in  45  B.  C.  which  ordered  that 


THE  CALENDAR  39 

the  calendar  year  should  consist  of  365  days  for  three  years  in  succession,  these 
to  be  followed  by  a  fourth  of  366  days.  The  extra  day  of  the  fourth  year  was 
introduced  by  counting  twice  the  sixth  day  before  the  calends  of  March  in  the 
Roman  system.  In  consequence  such  years  were  long  distinguished  by  the 
designation  bissextile,  although  they  are  now  called  Leap  Years.  The  years  of 
365  days  are  Common  Years.  With  this  arrangement  the  average  length  of 
the  calendar  year  was  365^  days.  This  period  is  called  a  Julian  Year,  and 
the  calendar  based  upon  it,  the  Julian  Calendar. 

The  difference  between  the  Julian  and  the  tropical  years  is  about  nm. 
In  order  to  avoid  the  gradual  displacement  of  the  calendar  dates  with  respect 
to  the  seasons  resulting  from  the  accumulation  of  this  difference,  a  slight  mod- 
ification in  the  method  of  counting  leap  years  was  introduced  in  1582  by  Pope 
Gregory  XIII.  The  accumulated  difference  amounts  approximately  to  three 
days  in  400  years,  and,  as  the  Julian  year  is  longer  than  the  tropical,  the  Julian 
calendar  falls  behind  the  seasons  by  this  amount.  Gregory  therefore  ordered  that 
the  century  years,  all  of  which  are  leap  years  under  the  Julian  rule,  should  not 
be  counted  as  such  unless  the  year  numbers  are  exactly  divisible  by  400.  At 
the  same  time  it  was  ordered  that  10  days  should  be  dropped  from  the  calendar 
in  order  to  bring  the  date  of  the  passage  of  the  sun  through  the  vernal  equinox 
back  to  the  2ist  of  March,  where  it  was  at  the  time  of  the  Council  of  Nice  in 
325  A.  D.  The  Julian  system  thus  modified  is  called  the  Gregorian  Calendar. 
The  revised  rule  for  the  determination  of  leap  years  is  as  follows :  All  years 
whose  numbers  are  exactly  divisible  by  four  are  leap  years,  excepting  the  century 
years.  These  are  leap  years  only  when  exactly  divisible  by  four  hundred.  All 
other  years  are  common  years.  The  average  length  of  the  Gregorian  calendar 
year  differs  from  that  of  the  tropical  year  by  only  0.0003  day  or  26s.  In  the 
modern  system  the  extra  day  in  leap  years  appears  as  the  2Qth  of  February. 

The  Gregorian  calendar  was  soon  adopted  by  all  Roman  Catholic  countries 
and  by  England  in  1752.  Russia  and  Greece  and  other  countries  under 
the  dominion  of  the  Eastern  or  Greek  Church,  still  use  the  Julian  Calendar,  which, 
at  present,  differs  from  the  Gregorian  by  13  days.  • 

24.  Given  the  local  time  at  any  point,  to  find  the  corresponding  local 
time  at  any  other  point. — From  the  definitions  of  apparent  solar,  mean  solar, 
and  sidereal  time,  it  follows  that  at  any  instant  the  difference  between  two  local 
times  is  equal  to  the  angular  distance  between  the  celestial  meridians  to 'which 
the  times  are  referred.  But  this  is  equal  to  the  angular  distance  between  the 
geographical  meridians  of  the  two  places,  i.e.  their  difference  of  longitude. 

Let  Te.  =  the  time  of  the  eastern  place, 
TV,  =  the  time  of  the  western  place, 
L  =  longitude  difference  of  the  two  places, 


We  then  have  the  relations: 

Te  =  Tw  +  L 


(41) 


40  PRACTICAL  ASTRONOMY 

Equations  (41)  are  true  whether  the  times  be  apparent  solar,  mean  solar,  or 
sidereal. 

Example  8.     Given,  Columbia  mean  solar  time  i2h  i4m  16541,  find  the  corresponding 
Greenwich  mean  solar  time. 

Tw=  i&  I4m  16541 
L  =  6  9  18.33 
Te  =  18  23  34.74  Ans. 

Example  9.     Given,  Greenwich  mean  solar  time  1907,  Oct.  6  3h  i4m  21',  find  the  corre- 
sponding Washington  mean  solar  time. 

Te  —  1907,  Oct.  6    3h  I4m  2i8 
L    =  5      8     16 

TV  =  1907,  Oct.  522       65     Ans. 

Example  10.     Given,  central  standard   time  1907,  Oct.  12  6hi8ho"  A.M.,  find  the  cor- 
responding Greenwich  mean  solar  time,  astronomical  and  civil. 

TV  —  1907,  Oct.  12     6h  i8m  o8  A.M. 

Oct.  ii  18     18    o  astronomical 
L    =                             600 

7e  =  1907,  Oct.  12    o     18    o  astronomical    \ 

Oct.  12    o     18    o  P.M.  civil         / 

Example  11.     Given,    central    standard    time    1907,  Oct.   n  oh  3m  16518  P.M.   find  the 
corresponding  Columbia  mean  solar  lime,  civil  and  astronomical. 

Te  =  1907,  Oct.  ii  oh     3m  16518  P.M. 

L    =  9     i8-33 

TV  =  I907»  Oct.  1 1  ii     53     57.85  A.M.  civil         j     ^^ 

Oct.  10  23     53     57.85  astronomical    / 

25.  Given  the  apparent  solar  time  at  any  place,  to  find  the  corre- 
sponding mean  solar  time,  and  vice  versa. — From  equation  (36),  t  =  6  —  «, 
and  the  definitions  of  mean  solar  and  apparent  solar  time,  we  find 

M.  S.  T.  =  0  —  R.  A.  of  M.  S., 
A.  S.  T.  =  0  —  R.  A.  of  A.  S. 

whence 

M.  S.  T.  —  A.  S.  T.  =  R.  A.  of  A.  S.  —  R.  A.  of  M.  S. 
The  difference 

E  =  M.  S.  T.  —  A.  S.  T.  (42) 

is  called  the  Equation  of  Time.  The  equation  of  time  varies  irregularly 
throughout  the  year,  its  maximum  absolute  value  being  about  i6m.  It  is  some- 
times positive,  and  sometimes  negative,  since  the  right  ascension  of  the  apparent 
sun  is  sometimes  smaller  and  sometimes  greater  than  that  of  the  mean  sun.  The 
right  ascension  of  the  apparent  sun  is  calculated  from  the  known  orbital  motion 
of  the  earth.  The  right  ascension  of  the  mean  sun  is  known  from  its  manner 


TIME  TRANSFORMATION  41 

of  definition.  This  data  suffices  for  .the  calculation  of  E,  whose  values  are  tabu- 
lated in  the  various  astronomical  ephemerides.  In  the  American  Ephemeris  they 
are  given  for  instants  of  Greenwich  apparent  noon  on  page  I  for  each  month, 
and  for  Greenwich  mean  noon,  on  page  II.  The  former  page  is  used  when 
apparent  time  is  converted  into  the  corresponding  mean  solar  time,  and  the 
latter  when  apparent  solar  time  is  to  be  found  from  a  given  mean  solar  time. 
The  algebraic  sign  of  E  is  not  given  in  the  American  Ephemeris,  but  the  column 
containing  its  values  is  headed  by  a  precept  which  indicates  whether  it  is  to  be 
added  to  or  subtracted  from  the  given  time.  Values  of  E  for  times  other  than 
Greenwich  apparent  noon  and  Greenwich  mean  noon  must  be  obtained  by  inter- 
polation. This  operation  is  facilitated  by  the  use  of  the  hourly  change  in  E 
printed  in  the  columns  headed  "Difference  for  I  Hour,"  which  immediately  fol- 
low those  containing  the  equation  of  time.  If  the  time  to  be  converted  refers 
to  a  meridian  other  than  that  of  Greenwich,  the  corresponding  Greenwich  time 
must  be  calculated  before  the  interpolation  is  made.  Note  that  for  each  date 
the  difference  of  the  right  ascension  of  the  apparent,  or  true,  sun  in  column  two 
of  page  II,  and  the  right  ascension  of  the  mean  sun  in  the  last  column  of  the 
same  page,  is  equal  to  the  corresponding  value  of  E,  in  accordance  with  the 
definition. 

Example  12.     Given,  Greenwich  apparent  solar  time  1907,  Oct.  15  2h  6m  12506,  find  the 
corresponding  Greenwich  mean  solar  time. 

E  for  Gr.  A.  N.  1907,  Oct.  15  13™  56575     (Eph.  p.  164) 

Change  in  E  during  2h  6m  12"  -(-   1.20 

E  (to  be  subtracted  from  A.  S.  T.)  13     57.95 

Gr.  A.  S.  T.     1907,  Oct.  15  26     12.06 

Gr.  M.  S.  T.     1907,  Oct.  15  i     52     14.11     Ans. 

Example  13.     Given,   Greenwich   mean  solar   time  1907,  Oct.  15  ih  52*"  14511,   find  the 
corresponding  Greenwich  apparant  solar  time. 

E  for  Gr.  M.  N.  1907,  Oct.  15  13™  56588     (Eph.  p.  165) 

Change  in  E  during  ih  52""  14"  -(-  1.07 

E  (to  be  added  to  M.  S.  T.)  13     57.95 

Gr.  M.  S.T.     1907,  Oct.  15  i     52     14.11 

Gr.  A.  S.  T.     1907,  Oct.  15  2       6     12.06    Ans. 

Example   14.     Given,    central   standard    time  1907,  Oct.  20  nh  i8m  1252  A.M.,  find  the 
corresponding  Columbia  apparent  solar  time. 

C.  S.  T.                        1907,  Oct.  20  nh  i8m  1212      A.M. 

L  9     18.3 

Columbia  M.  S.  T.     1907,  Oct.  19  23       8     53.9       astronomical 

Gr.  M.  S.  T.                1907,  Oct,  20  5     18     12.2 

E  for  Gr.  M.  N.          1907,  Oct.  20  14     58.63     (Eph.  p.  165) 

Change  in  E  during  5h  i8m  12"  -(-  2.39 

E  (to  be  added  to  Columbia  M.  S.  T.)   15       i.o 

Columbia  A.  S.  T.     1907,  Oct.  19  23     23     54.9       Ans. 


42  PRACTICAL  ASTRONOMY 

Example  15.     Given  Columbia  apparent  solar  time  1907,  Oct.  19  23^  23™  5419,  find  the 
corresponding  central  standard  time. 

Columbia  A.  S.  T.     1907,  Oct.  19     23h  23'"  5419 
L  69     18.3 

Gr.  A.  S.  T.  1907,  Oct.  20       5     33     13.2 

E  for  Gr.  A.  N.  1907,  Ot't.  20  14    58.52 

Change  in  £  during  5h33m  i3B  _|_  2.50 

E  (to  be  sub.  from  Columbia  A.  S.  T.)     15       i.o 

Columbia  M.  S.  T.    1907,  Oct.  19     23      8     53.9 

L  9     18.3 

C.  s-  T.  1907,  Oct.  20     ii     18     12.2       A.M.     Ans. 

26.  Relation  between  the  values  of  a  time  interval  expressed  in 
mean  solar  and  sidereal  units.  Equation  (40)  is  the  fundamental  relation 
connecting  the  units  of  mean  solar  and  sidereal  time.  If  we  let 


/s   =  the  value  of  any  interval  /in  mean  solar  units, 
7m  =  the  value  of  /in  sidereal  units, 


we  find  from  (40) 


/s    _  /m  +  (43) 


/S 

(44) 


Writing 


HI  =  r^r-—  II  = 


365.2422  366.2422 

(43)  and  (44)  become 

/s   =  /m  +  in/m 

/m  =  /s  —    H/s  (46) 

Assuming  /m  =  24h  we  find  from  (45) 

24h  om  o'ooo  M.  S.  =  24h  3m  56:555  Sid. 
Similarly,  by  supposing  /s  =  24h  we  obtain  from  (46) 

24h  om  o?ooo  Sid.  =  23h  56m  4^091  M.  S. 
Hence 

Gain  of  0  on  M.  S.  T.  in  I  M.  S.  D.  =  IIl24h  =  2365555 

Gain  of  6  on  M.  S.  T.  in  i        S.  D.  =    1124    =  235.909        '47> 


RELATION  OF  SIDEREAL  AND  SOLAR  INTERVALS  43 

and  further 

Gain  of  0  on  M.  S.  T.  in  I  M.  S.  hour  =  IIIi»  =  958565 
Gainof0onM.S.T.ini        S.  hour  =    Hi    =9.8296 

For  many  purposes  these  expressions  may  be  replaced  by  the  following  approx- 
imate relations: 

IIl24h  —  4m(i  —  1/70),  Error  =  0*016 

1124  =  4  (i  —  1/60),  Error  =  0.081 

IIIi   =IO(I  —  1/70),  Error  =  0.0006 

Hi  =10  (i  —  1/60),  Error  —  0.0037 

Equations  (45)  and  (46)  may  be  used  for  the  conversion  of  the  value 
of  a  time  interval  expressed  in  mean  solar  units  into  its  corresponding  value 
in  sidereal  units,  and  vice  versa.  The  calculations  are  most  conveniently  made 
by  Tables  II  and  III  printed  at  the  end  of  the  American  Ephemeris.  Table  II 
contains  the  numerical  values  of  II/s,  while  Table  III  gives  those  of  III/m,  the 
arguments  being  the  values  of  /s  and  7m,  respectively.  It  will  be  observed  that 
the  first  factors  of  II/s  and  III/m  indicate  the  table,  and  the  second  the  argu- 
ment which  is  to  be  used  for  the  interpolation. 

In  case  tables  are  not  available  the  conversion  can  be  based  upon  equations 
(47)  or  (48),  or  more  simply,  upon  (49),  provided  the  highest  precision  is  not 
required. 

Example  16.     Given  the  mean  solar  interval  i6hi8m2i!2o,   find  the  equivalent  sidereal 
interval. 

By  Eq.  (45)  7m  =  i6h  *l8m  21520 

Ill/m  =  2     40.72     (Eph.  Table  III) 

/s  =  16     21       1.92     Ans. 

The  calculation  of  III/m  by  the  third  of  (49)  is  as  follows: 

/m  =  i6'.'3o6        ios/m  =  i63fo6 
i/7oXio'/m=      2.33 

Ill/m  =  160.73  =  2m  4°573 
The  value  thus  found  differs  onlyo'oi  from  that  derived  from  Table  III  of  the  Ephemeris. 

Example  17.     Given  the  sidereal  interval  2oh28m  42517,   find  the  equivalent  mean  solar 
interval. 

By  Eq.  (46)  7S  =  2Oh  28™  42517 

II/s  =  3     21.29     (EPh-  T»ble  II) 

/m  =  20     25     2O.88     Ann. 

The  calculation  of  II/S  by  the  last  of  (49)  is  as  follows: 

/s  =  20^478         io8/s  =  204578 
i/6oXio9/s  =      3.41 

II/s  =  201.37  =  3m  21537 


44  PRACTICAL  ASTRONOMY 

27.  Relation  between  mean  solar  time  and  the  corresponding  side- 
real time.  —  In  Section  14  it  was  shown  that  the  relation  connecting  the  hour 
angle  of  an  object  with  the  sidereal  time  is 


where  a  represents  the  right  ascension  of  the  object.  Applying  this  equation 
to  the  mean  sun,  we  find 

M  =  e  —  R  (50) 

in  which  R  represents  the  right  ascension  of  the  mean  sun,  and  M  its  hour 
angle.  The  latter,  however,  is  equal  by  definition  to  the  mean  solar  time.  Equa- 
tion (50)  therefore  expresses  a  relation  between  mean  solar  time  and  the  cor- 
responding sidereal  time,  which  can  be  made  the  basis  for  the  conversion  of 
the  one  into  the  other.  The  transformation  requires  a  knowledge  of  R,  the 
right  ascension  of  the  mean  sun,  at  the  instant  to  which  the  given  time  refers. 
We  now  turn  our  attention  to  a  consideration  of  the  methods  which  are  avail- 
able for  the  determination  of  this  quantity. 

28.    The  right  ascension  of  the  mean  sun  and  its  determination.  —  It 

is  shown  in  works  on  theoretical  astronomy  that  the  right  ascension  of  the 
mean  sun  at  any  instant  of  Greenwich  mean  time  is  given  by  the  expression 

^G  =  i8h  38m  45*836  +  (236:555  X  365.25)' 

+  of  0000093  'a  (51) 

+  nutation  in  right  ascension, 

in  which  t  is  reckoned  in  Julian  years  from  the  epoch  1900,  Jan.  od  oh  Gr.  M.  T. 
It  thus  appears  that  the  increase  in  the  right  ascension  of  the  mean  sun  is  not 
strictly  proportional  to  the  increase  in  the  time.  This,  in  connection  with  equation 
(50),  shows  that  sidereal  time  is  not  a  uniformly  varying  quantity,  a  fact  already 
indicated  in  Section  21.  The  nutation  in  right  ascension  oscillates  between  limits 
which  are  approximately-}-  Is  and  —  Is  with  a  period  of  about  19  years.  Its  change 
in  one  day  is  therefore  very  small,  and,  as  the  same  is  true  of  the  term  involving 
t2  in  (51),  it  follows  that  the  increase  in  the  right  ascension  of  the  mean  sun 
in  one  mean  solar  day  is  sensibly  236.S555.  From  equation  (50)  it  is  seen  that 
the  gain  of  sidereal  on  mean  solar  time  during  any  interval  is  equal  to  the 
increase  in  R  during  that  interval;  and,  indeed,  we  have  exact  numerical  agree- 
ment between  the  change  in  the  latter  for  one  mean  solar  day,  as  given  by  equa- 
tion (51),  and  the  gain  of  the  former  during  the  same  period  as  shown  by  the 
first  of  (47).  From  this  it  follows  that  the  methods  given  in  Section  26,  including 
Tables  II  and  III  of  the  Ephemeris  and  the  approximate  relations  (49),  can 
equally  well  be  applied  to  the  determination  of  the  increase  in  R,  provided  only 
that  the  interval  for  which  the  cKange  is  to  be  calculated  is  small  enough  to 
render  the  variations  in  the  last  two  terms  of  (51)  negligible. 


RIGHT  ASCENSION  OF  MEAN  SUN  45 

To  facilitate  the  solution  of  problems  in  which  R  is  required,  its  precise 
numerical  values  are  tabulated  in  the  various  astronomical  ephemerides  for  every 
day  in  the  year.  In  the  American  Ephetnieris  they  are  given  for  the  instant  of 
Greenwich  mean  noon,  and  are  to  be  found  in  the  last  column  of  page  II  for 
each  month.  If  these  tabular  values  be  represented  by  Ro,  and  if  J?L  represent 
the  right  ascension  of  the  mean  sun  at  the  instant  of  mean  noon  for  a  point  whose 
longitude  west  of  Greenwich  is  L,  it  follows  from  the  preceding  paragraph 
that 

RL  =  Ro  +  IIIZ,  (52) 

for  L  is  equal  to  the  time  interval  separating  mean  noon  of  the  place  from  the 
preceding  Greenwich  mean  noon.  Further,  the  value  of  R  at  any  mean  time,  M, 
at  a  point  whose  longitude  west  of  Greenwich  is  L  is  given  by 

R  =  XL  +  II W,  (53) 

or 

R  =  Ro  +  IIIZ  +  HIM.  (54) 

Equations  (52)  and  (53),  or  their  equivalent,  (54),  suffice  for  the  determi- 
nation of  R  at  any  instant  at  any  place  when  the  value  of  Ro  for  the  preceding 
mean  noon  is  known.  For  a  given  place  the  term  lllL  is  a  constant.  Its  value 
can  be  calculated  once  for  all,  and  can  then  be  added  mentally  to  the  value  of 
Ro  as  the  latter  is  taken  from  the  Ephemeris.  The  quantity  HIM  may  be 
derived  from  Table  III  of  the  Ephemeris  with  M  as  argument. 

If  an  Ephemeris  is  not  available  the  values  of  R  can  still  be  found;  approxi- 
mately at  least,  by  the  use  of  Tables  II-IV,  page  46.  The  first  of  these  contains 
the  values  of  Ro  computed  from  (51)  for  the  date  Jan.  o  for  each  of  the  years 
1907-1920.  Denoting  these  by  Roo  and  neglecting  the  variations  in  the  last  two 
terms  of  (51)  we  have  for  Greenwich  mean  noon  of  any  other  date 

Ro  =  Roo  -f-  IIIZ?  (55) 

where  D  indicates  the  number  of  mean  solar  days  that  have  elapsed  since  the 
preceding  Jan.  o.  Substituting  (55)  into  (54). 

R  =  R00  +  IIIZ  +  lll(D  +  M).  (56) 

The  value  of  D  may  be  obtained  from  Table  III  by  adding  the  day  of  the 
month  to  the  tabular  number  standing  opposite  the  name  of  the  month  in 
question.  M  is  conveniently  expressed  in  decimals  of  a  day  by  means  of  Table 
IV.  The  value  thus  found  is  to  be  combined  with  D.  If  the  precise  value  of 
III,  viz.,  236.S555,  be  used,  the  uncertainty  in  R  derived  from  (56)  will  be  only 
that  arising  from  the  neglect  of  the  variation  in  the  last  terms  of  (51).  If  care 
be  taken  to  count  D  from  the  nearest  Jan.  o  the  error  will  never  exceed  o.83  or 


46 


PRACTICAL  ASTRONOMY 


TABLE  II 

RIGHT  ASCENSION  OF  THE  MEAN  SUN  FOR  THE 
EPOCH  JAN.  od  oh  GR.  M.  T. 


Year 

ROO        «* 

Year 

R 

oo 

1907 

i8h  36™ 

0547 

1914 

i8h  371 

"  13562 

1908 

35 

3-04 

1915 

36 

16.64 

1909 

38 

2.28 

1916 

35 

19.63 

1910 

37 

5-09 

1917 

38 

1907 

1911 

36 

7-99 

1918 

37 

21.85 

1912 

35 

10.98 

1919 

36 

24.51 

1913 

38 

10.57 

1920 

35 

28.05 

Date 

D 

Common 
Year 

Leap 
Year 

Jan.     o 

o 

o 

Feb.    o 

31 

31 

Mar.    o 

59 

60 

Apr.    o 

90 

9i 

May    o 

1  20 

121 

June    o 

151 

152 

July    o 

isn 

l82l 

Aug.   o 

2I2\ 
—  153  J 

213) 
—  153  J 

Sept.  o 

243  \ 
—  122  / 

244  \ 
122  j 

Oct.    o 

2731 
—    92/ 

2741 
—     92  / 

Nov.    o 

304  1 

3051 

Dec.    o 

334  1 

335  j 

PRECEPT:  Add  the  day  of 
the  month  to  the  tabular  value 
corresponding  to  the  given 
month.  The  use  of  the  nega- 
tive values  gives  the  day  num- 
ber from  Vnz  following  Jan.  o. 


Decimals 

Decimals 

Decimals 

Hour 

of  a 

Min. 

of  a 

Min. 

of  a 

Day 

Day 

Day 

I 

0.042 

i 

O.OOI 

10 

0.007 

2 

0.083 

2 

I 

20 

0.014 

3 

0.125 

3 

2 

30 

O.02I 

4 

0.167 

4 

3 

40 

0.02S 

5 

0.208 

5 

3 

50 

0.035 

6 

0.250 

6 

4 

60 

O.O42 

7 

0.292 

7 

5 

8 

0-333 

8 

6 

9 

0-375 

9 

6 

10 

0.417 

10 

0.007 

ii 

0.458 

12 

0.500 

PRECEPT:  When  the  given  hour  is  greater  than 
12,  drop  i2h  from  the  argument  and  add  od5oo  to  the 
result  given  by  the  table.  Thus,  for  I7h28m  enter  the 
table  with  the  argument  5h28m,  giving  od228,  whence 
i7h  28m  =  od228  -j-  od50o  =  od728. 


TRANSFORMATION  OF  MEAN  SOLAR  INTO  SIDEREAL  TIME  47 

0*4.     This  requires  that  for  D>  i83d  the  negative  value  of  Table  III  be  employed, 
together  with  the  value  of  Roo  for  ihe  following  Jan.  o. 

If  a  somewhat  greater  uncertainty  is  permissible,  the  result  may  be  more 
expeditiously  found  by  using  4m(i  —  1/70)  for  III.  If  D  be  reckoned  from  the 
nearest  Jan.  o  as  above,  the  corresponding  error  will  not  exceed  3s. 

Example  18.     Find  the  right  ascension  of  the   mean    sun  for  the  epoch   1907,  June  16 
8h  2im  14500  Columbia  M.  S.  T. 

By  Equation  (54) 

Xo  =  5h  34m  25510  (Eph.  p.  93) 

L  =  6h     9m  18133         11IL  =  i       0.67  (Eph.  Table  III) 

Af=8     21     14.00       IIU/=          i     22.34  (Eph.  Table  III) 

R  =  5     36     48.11  Ans. 

By  Equation  (56) 

(D  +  M)  =  i67d348  (Tables  III  and  IV;  R00  (1907)  =  i8h  36m    055     (Table  II) 
4m  (D  +  M*)  =  669T392  IIIZ,  =  i       0.7 

(^  +  ^)  =       91-563  lll(D  +  M)  =  io    59     49.7 

^=5    36    51 


Example  19.     Find  the  right  ascension   of  the  mean  sun  for  the  epoch  1909,  Sept.  21 
I9h  26m  248  Columbia  M.  S.  T. 


=  —  101  d  +  od8io  (Tables  III  and  IV)Jff0o  (1910)  =    i8h  37™  5*1  (Table  II) 

=  —  100^190  III/,—  i     0.7 

4m  (D+M)  =  —  4001760  III(Z?  -f  M)  =  —  6  35     2.1 

i/7oX4m(£>+Af)  =—      5.725  R=    12  34       Ans. 

The  precise  value  given  by  (54)  is  I2h3m5f2i. 

29.  Given  the  mean  solar  time  at  any  instant  to  find  the  corre- 
sponding sidereal  time.  —  From  equation  (50)  we  find 

d  =  M+R  (57) 

Introducing  the  value  of  R  from  (53)  we  have 

6  =  M+£i.  +  lUM,  (58) 

where 

£L  =  R0  +  HIZ.  (59) 

Equations  (59)  and  (58)  solve  the  problem. 

Equation  (58)  may  be  interpreted  as  follows:  RL  is  the  right  ascension: 
of  the  mean  sun  at  the  preceding  mean  noon  for  a  place  in  longitude  L  west  of 
Greenwich.  It  is  therefore  also  equal  to  the  hour  angle  of  the  vernal  equinox 
at  that  instant,  i.e.  to  the  sidereal  time  of  the  preceding  mean  noon  at  the 
place  considered.  Now  M  is  the  mean  time  interval  since  preceding  mean  noon, 
and  by  (45)  Af-f-IIIM  is  the  equivalent  sidereal  interval.  The  right  member 
of  (58)  therefore  expresses  the  sum  of  the  sidereal  time  of  the  preceding  mean 
noon  and  the  number  of  sidereal  hours,  minutes,  and  seconds  that  have  elapsed 


48  PRACTICAL  ASTRONOMY 

since  noon.  In  other  words  it  is  the  sidereal  time  corresponding  to  the  mean 
time,  M}  as  indicated  by  the  equation. 

In  case  the  Ephemeris  is  not  at  hand,  R  may  be  obtained  from  (56)  and 
substituted  into  (57)  for  the  determination  of  6.  The  uncertainty  in  the  sidereal 
time  thus  found  will  be  the  same  as  th#t  of  R  derived  from  (56). 

Oftentimes  a  rough  approximation  for  6  is  all  that  is  required.  In  such 
cases  the  following,  designed  for  use  at  the  meridian  of  Columbia,  is  useful : 

0  =  i8h  3717  -f  M  +  4m(i  —  1/70)  (D  +  M).  (60) 

The  first  term  in  the  right  member  of  this  formula  is  the  average  value  of 
Roo  plus  the  constant  term  IIIL,  which  for  Columbia  may  be  taken  equal  to  im. 
The  expression  can  be  adapted  for  use  at  any  other  meridian  by  introducing  the 
appropriate  value  of  IIIL.  The  maximum  error  in  the  value  of  6  derived  from 
(60)  is  i.m7. 

Example  20.     Given  Columbia  mean  solar  time  i6h27m  32517  on  1909,  Nov.  16,  find  the 
corresponding  sidereal  time. 

By  equations  (58)  and  (59) 
M '  =  i6h  27m  32517 

Ro  =  15  39  39-98 

IIIL  =  i  0.67 

IIIM=  2  42.23 

S  =    8  10  55.05     Ans. 

By  equations  (56)  and  (57) 

D  +  M  =  —  45d  +  od686  M =    i6h  27™  3252 

=  —  44d3H  -ffoo=    18     37      5.1 

4m(Z>  +  ^7)  =—  177^256  I1IZ,  =  i       0.7 

i/70X4m(Z>-f^/)  =  -       2.532  lll(D  +  M)=—  2     54     43.4 

B  =      8     10     55        Ans. 

By  equation  (60) 

M=    16     27.5 
4m(i  —  1/70)  (Z>  +  M)  =  — 2     54.7 

0  =      8     10. 5     Ans. 

30.  Given  the  sidereal  time  at  any  instant  to  find  the  corresponding 
mean  solar  time. — We  make  use  of  equation  (50),  viz. 


Substituting  as  in  Section  29  we  have 

M=  6  —  XL—HIM 

or 

d  —  RL  (61) 


49 


Multiplying  equations    (45)    and    (46),  member  by  member,  and  dropping  the 
common  factor  7m  7s  we  find 


(i  +III)(i  -II)-  i 
Combining  this  with  (61)  we  find 

M  =  6  —  RL  —  11(0  —  XL),  (62) 

where,  as  before, 

RL  =  R0  -f  III7  (63) 

Equations  (63)  and  (62)  solve  the  problem. 

Equation  (62)  is  susceptible  of  an  interpretation  similar  to  that  given  (58) 
in  the  preceding  section.  Since  6  is  the  given  sidereal  time,  and  R\.  the  sidereal 
time  of  the  preceding  mean  noon,  6  —  T^L  is  the  sidereal  interval  that  has  elapsed 
since  noon.  To  find  the  equivalent  mean  time  interval  we  must,  in  accordance 
with  equation  (46),  subtract  from  6  —  RL  the  quantity  1  1(0  —  RL).  The  right 
member  of  (62)  therefore  expresses  the  number  of  mean  solar  hours,  minutes, 
and  seconds  that  have  elapsed  since  the  preceding  mean  noon,  i.e.  the  mean 
solar  time  corresponding  to  the  given  6. 

Example  21.     Given,  1908,  May   12,  Columbia  sidereal  time  ih7m  19*27,  find  the  corre- 
sponding central  standard  time. 

By  equations  (62)  and  (63) 

S  =    ih     7m  19128 

^?L=    3     20     25.46 

9  —  #L=2i     46     53.82 

11(0  —  7?L)=  3     34-io     (Eph.  Table  II) 

M  =  21     43     19.72 
Z,  =  9     18.33 

C.  S.  T.  =    9     52     38.05     A.M.  May  13.     Ans. 


CHAPTER    IV 

INSTRUMENTS  AND  THEIR  USE 

31.  Instruments  used  by  the  engineer. — The  instruments  employed  by  the 
engineer  for  the  determination  of  latitude,  time  and  azimuth  are  the  watch  or 
chronometer,  the  artificial  horizon,  arid  the  engineer's  transit  or  the  sextant.     The 
following  pages  give  a  brief  account  of  the  theory  of  these  instruments  and  a 
statement  of  the  methods  to  be  followed  in  using  them. 

The  use  of  both  the  engineer's  transit  and  the  sextant  presupposes  an  under- 
standing of  the  vernier.  In  consequence,  the  construction  and  theory  of  this  at- 
tachment is  treated  separately  before  the  discussion  of  the  transit  and  sextant 
is  undertaken. 

TIMEPIECES 

32.  Historical. — Contrivances  for  the  measurement  of  time  have  been  used 
since  the  beginning  of  civilization,  but  it  was  not  until  the  end  of  the  sixteenth 
century  that  they  reached  the  degree  of  perfection  which  made  them  of  service 
in  astronomical  observations.     The  pendulum  seems  first  to  have  been  used  as  a 
means  of  governing  the  motion  of  a  clock  by   Biirgi   of  the  observatory  of 
Landgrave  William  IV  at  Cassel  about  1580,  though  it  is  not  certain  that  the 
principle  employed  was  that  involved  in  the  modern  method  of  regulation.    How- 
ever this  may  be,  the  method  now  used  was  certainly  suggested  by  Galileo  about 
1637;  but  Galileo  was  then  near  the  end  of  his  life,  blind  and  enfeebled,  and  it 
was  not  until  some  years  later  that  his  idea  found  material  realization  in  a  clock 
constructed  by  his  son  Vincenzio.   It  remained  for  Huygens,  however,  the  Dutch 
physicist  and  astronomer,  to  rediscover  the  principle,  and  in  1657  give  it  an  appli- 
cation that  attracted  general  attention.     Some  sixty  years  later  Harrison  and 
Graham  devised  methods  of  pendulum  compensation  for  changes  of  temperature, 
which,  with  important  modifications  in  the  escapement  mechanism    introduced  by 
Graham  in  1713,  made  the  clock  an  instrument  of  precision.     Since  then  its  devel- 
opment in  design  and  construction  has  kept  pace  with  that  of  other  forms  of 
astronomical  apparatus. 

The  pendulum  clock  must  be  mounted  in  a  fixed  position.  It  can  not 
be  transported  from  place  to  place,  and  it  does  not,  therefore,  fulfill  all  the 
requirements  that  may  be  demanded  of  a  timepiece.  By  the  beginning  of  the 
eighteenth  century  the  need  of  accurate  portable  timepieces  had  become  pressing, 
not  so  much  for  the  work  of  the  astronomer  as  for  that  of  the  navigator.  The 
most  difficult  thing  in  finding  the  position  of  a  ship  is  the  determination  of  longi- 
tude. At  that  time  no  method  was  known  capable  of  giving  this  with  anything 
more  than  the  roughest  approximation,  although  the  question  had  been  attacked 
by  the  most  capable  minds  of  the  two  centuries  immediately  preceding.  The 
matter  was  of  such  importance  that  the  governments  of  Spain,  France,  and  the 
Netherlands  established  large  money  prizes  for  a  successful  solution,  and  in  1714 
that  of  Great  Britain  offered  a  reward  of  £20,000  for  a  method  which  would  give 
the  longitude  of  a  ship  within  half  a  degree.  With  an  accurate  portable  timepiece, 

50 


TIMEPIECES  51 

which  conlcl  be  set  to  indicate  the  time  of  some  standard  meridian  before  begin- 
ning a  voyage,  the  solution  would  have  been  simple.  Notwithstanding  the  stimu- 
lus of  reward  no  solution  was  forthcoming  for  many  years.  In  1735  Harrison 
succeeded  in  constructing  a  chronometer  which  was  compensated  for  changes  of 
temperature;  and  about  1760  one  of  his  instruments  was  sent  on  a  trial  voyage 
to  Jamaica.  Upon  return  its  variation  was  found  to  be  such  as  to  bring  the 
values  of  the  longitudes  based  on  its  readings  within  the  permissible  limit  of 
error. 

The  ideal  timepiece,  so  far  as  uniformity  is  concerned,  would  be  a  body  moving 
under  the  action  of  no  forces,  but  in  practice  this  can  not  be  realized.  The 
modern  timepiece  of  precision  is  a  close  approximation  to  something  equivalent, 
but  falls  short  of  the  ideal.  Thus  far  it  has  been  impossible  completely  to  nullify 
the  effect  of  certain  influences  which  affect  the  uniformity  of  motion.  Changes 
in  temperature,  variations  in  barometric  pressure,  and  the  gradual  thickening  of 
the  oil  lubricating  the  mechanism  produce  irregularities,  even  when  the  skill  of 
the  designer  and  clockmaker  is  exercised  to  its  utmost.  No  timepiece  is  perfect. 
We  can  say  only  that  some  are  better  than  others.  Further,  it  is  impossible 
to  set  a  timepiece  with  such  exactness  that  it  does  not  differ  from  the  true  time 
by  a  quantity  greater  than  the  uncertainty  with  which  the  latter  can  be  determined. 
Thus  it  happens  that  a  timepiece  seldom  if  ever  indicates  the  true  time;  and,  in 
general,  no  attempt  is  made  to  remove  the  error.  The  timepiece  is  started  under 
conditions  as  favorable  as  possible,  and  set  to  indicate  approximately  the  true 
time.  It  is  then  left  to  run  as  it  will,  the  astronomer,  in  the  meantime,  directing 
his  attention  to  a  precise  determination  of  the  amount  and  the  rate  of  change  of 
the  error.  These  being  known,  the  true  time  at  any  instant  is  easily  found. 

33.  Error  and  rate. — The  error,  or  correction,  of  a  timepiece  is  the  quantity 
which  added  algebraically  to  the  indicated  time  gives  the  true  lime.  The  error 
of  a  timepiece  which  is  slow  is  therefore  positive.  If  the  timepiece  is  fast  the 
algebraic  sign  of  its  correction  is  negative. 

The  error  of  a  mean  solar  timepiece  is  denoted  by  the  symbol  JT;  of  a 
sidereal  timepiece,  by  Jt).  To  designate  the  timepiece  to  which  the  correction 
refers  subscripts  may  be  added.  Thus  the  error  of  a  Fauth  sidereal  clock  may 
be  indicated  by  J6f;  of  a  Negus  mean  time  chronometer,  by  JT^.  Sometimes 
it  is  convenient  to  use  the  number  of  the  timepiece  as  subscript. 

If  6'  be  the  indicated  sidereal  time  at  a  given  instant,  and  JO  the  cor- 
responding error  of  the  timepiece,  the  true  time  of  the  instant  will  be 

e  =  d'-\-Jd'.  (64) 

The  analogous  formula  for  a  mean  solar  timepiece  is 

T=  T  +  JT.  (65) 

The  daily  rate,  or  simply  the  rate,  of  a  timepiece  is  the  change  in  the  error 
during  one  day. 


52  PRACTICAL  ASTRONOMY 

If  the  error  of  a  timepiece  increases  algebraically,  the  rate  is  positive;  if  it 
decreases,  the  rate  is  negative.  The  symbols  od  and  oT  with  appropriate  sub- 
scripts are  used  for  the  designation  of  the  rates  of  sidereal  and  mean  solar  time- 
pieces, respectively.  The  hourly  rate,/.^.  the  change  during  one  hour,  is  some- 
times more  conveniently  employed  than  the  daily  rate. 

It  is  convenient,  but  in  no  wise  important,  that  the  rate  of  a  timepiece  should 
be  small.  On  the  other  hand,  it  is  of  the  utmost  consequence  that  the  rate  should 
be  constant;  for  the  reliability  of  the  instrument  depends  wholly  upon  the  degree 
to  which  this  condition  is  fulfilled. 

Generally  it  is  impossible  to  determine  by  observation  the  error  at  the  instant 
for  which  the  true  time  is  required.  We  must  therefore  be  able  to  calculate  its 
value  for  the  instant  in  question  from  values  previously  observed.  If  the  rate 
is  constant  this  can  be  done  with  precision;  otherwise,  the  result  will  be  affected 
by  an  uncertainty  which  will  be  the  greater,  the  longer  is  the  interval  separating 
the  epochs  of  the  observed  and  the  calculated  errors. 

If  J#  and  JO'  be  values  of  the  observed  error  for  the  epochs  t  and  /',  the 
daily  rate  will  be  given  by 

»»=^  <66> 


in  which  t'  —  /must  be  expressed  in  days  and  fractions  of  a  day.  The  rate  having 
thus  been  found,  the  error  for  any  other  epoch,  /",  may  be  calculated  by  the 
formula 

Jd"  =  J6'  +  dd(t"  —  t')  (67) 

Example  22.  The  error  of  a  sidereal  clock  was  -f-  5™  27161  on  1909,  Feb.  3,  at  6'.'4 
sidereal  time,  and  +  5m  33!io  on  1909,  Feb.  11,  at  5'.'2;  find  the  daily  rate,  and  the  correction  on 
Feb.  14  at  7^6  sidereal  time. 

We  have  J0  =  +  5m  27*61,  J0'  --=  +  5m  33!  10,  and 

t'  —  t  =  i  id  5^2  —  3d  61'4  —  7d  22V8  =  7^95. 

Equation  (66)  then  gives  80  =  +  5H9/7-9S  =  4  0569,  which  is  the  required  value  of  the  rate. 
To  find  the  error  for  Feb.  14,  7^6,  we  have 

t"  —  t1  —  i4d  7>?6  —  i  id  5'.'2  =  3d  2l'4  =  3<?i, 
whence  by  equation  (67) 

A0"=  +  5m  SS^o  +  S-iX  0169  =  +  5m  35124.     A»s. 

34.  Comparison  of  timepieces.  —  It  is  frequently  necessary  to  know  the  time 
indicated  by  one  timepiece  corresponding  to  that  shown  by  another.  The  determi- 
nation of  such  a  pair  of  corresponding  readings  involves  a  comparison  of  the  two 
timepieces.  To  make  such  a  comparison  the  observer  must  be  able  accurately 
to  follow,  or  count,  the  seconds  of  a  timepiece  without  looking  at  the  instrument. 
It  is  desirable,  moreover,  that  he  should  be  able  to  do  this  while  engaged  with 
other  matters,  such  as  entering  a  record  in  the  observing  book,  etc. 


CLOCK  COMPARISONS  53 

Pendulum  clocks  usually  beat,  or  tick,  every  second ;  and  chronometers,  every 
half  second.  The  beats  of  the  ordinary  watch  are  separated  by  an  interval  of  a 
fifth  of  a  second.  With  each  beat  the  second  hand  of  the  timepiece  moves 
forward  by  an  amount  corresponding  to  the  interval  separating  the  beats — a 
whole  second  space  for  the  clock,  a  half  second  space  for  the  chronometer,  and 
a  fifth  of  a  second  for  the  watch. 

If  the  beats  of  two  timepieces  coincide,  a  comparison  is  easily  made.  The 
observer  has  only  to  pick  up  the  beat  from  one,  then,  following  mentally,  look 
at  the  other  and  note  the  hour,  minute,  and  second  corresponding  to  a  definite 
time  on  the  first.  After  noting  the  reading  of  the  second,  the  observer  should 
look  again  at  the  first  before  dropping  the  count,  to  make  sure  that  the  indicated 
number  of  seconds  and  the  count  were  in  agreement  at  the  instant  of  comparison 
If  the  beats  of  the  timepieces  do  not  coincide,  and  it  is  desired  to  obtain  a  com- 
parison with  an  uncertainty  less  than  the  beat  interval,  the  observer  must  estimate 
from  the  sound  the  magnitude  of  the  interval  separating  the  ticks.  He  will  then 
note  the  hour,  minute,  second,  and  tenth  of  a  second  on  the  second  timepiece  cor- 
responding to  the  beginning  of  a  second  on  the  first. 

When  a  watch  is  to  be  compared  with  a  clock  or  a  chronometer,  the  count 
should  be  taken  from  the  latter.  The  tenths  of  a  second  on  a  watch  corresponding 
to  the  beginning;  of  a  second  on  the  clock  or  chronometer  may  be  estimated  by 
noting  the  position  of  the  watch  second  hand  with  respect  to  the  two  adjacent 
second  marks  at  the  instant  the  beat  of  the  clock  or  chronometer  occurs.  The 
comparison  will  then  give  the  hour,  minute,  second,  and  zero  tenths  on  the  clock 
or  chronometer  corresponding  to  a  certain  hour,  minute,  second,  and  tenth  of  a 
second  on  the  watch. 

If  a  sidereal  and  a  mean  solar  timepiece  are  to  be  compared,  a  very  precise 
result  may  be  obtained  by  the  method  of  coincident  beats.  It  was  shown  in  Sec- 
tion 26  that  the  gain  of  sidereal  on  mean  solar  time  is  about  ten  seconds  per  hour, 
or  one  second  in  six  minutes.  The  ticks  of  a  solar  and  a  sidereal  timepiece,  each 
beating  seconds,  must  therefore  coincide  once  every  six  minutes.  If  one  of  the 
timepieces  beats  half  seconds,  the  coincidences  will  occur  at  intervals  of  three  min- 
utes. A  comparison  is  made  by  noting  the  times  indicated  by  the  two  instruments 
at  the  instant  the  beats  coincide.  If  carefully  made,  the  uncertainty  of  the  com- 
parison will  not  exceed  one  or  two  hundredths  of  a  second. 

Example  23.  On  1907,  Oct.  29,  five  comparisons  of  a  watch  were  made  with  the  Fauth 
sidereal  clock  of  the  Laws  Observatory.  The  means  of  the  comparisons  are  OF  =  i8h  23™  ofoo, 
and  7\v  =  4h3m  16112  P.M.  The  error  of  the  Fauth  clock  was — 29572,  and  the  longitude 
west  of  Greenwich  is  6h  9'"  18*33.  Find  the  error  of  the  watch  referred  to  central  standard 
time. 

From  OF  and  J#F  find  ()  by  (64).  The  sidereal  time  is  then  to  be  transformed  into  C.  S.  T. 
by  (62)  and  the  first  of  (41;.  The  resulting  C.  S.  T.  compared  with  7\v  gives  the  error  of 
the  watch. 


54  PRACTICAL  ASTRONOMY 


9r 

iSh 

23m 

o?oo 

J0F 

— 

29.72 

0 

18 

22 

30.28 

Rl. 

H 

27 

40.63 

0  —  7?L 

3 

54 

49-65 

11(0  —  /?,.) 

. 

38.47 

Col.  M.  S.  T 

•'    3 

54 

ii.  18 

Z, 

9 

J8-33 

C.  S.  T. 

4 

3 

29.51     P.  M. 

TV 

4 

3 

16.12     P.M. 

J7\v 

+ 

13.39     -Ans. 

Example  24.  On  1907,  Oct.  30,  civil  date,  the  Fauth  sidereal  clock  of  the  Laws  Observ- 
atory read  I4h  28™  9175,  when  the  Riggs  clock,  a  central  standard  timepiece,  indicated 
oh  ^m  J^BOQ  P.M.  The  error  of  Riggs  was  +4582;  find  the  correction  to  the  Fauth  clock. 

The  reading  of  Riggs  combined  with  its  error  by  (65)  gives  the  true  C.  S.  T.  From  this 
the  Columbia  M.  S.  T.  is  found  by  the  second  of  (41).  This  converted  into  the  correspond- 
ing 0  by  (58)  and  compared  with  the  reading  of  Fauth  gives  J#F. 

In  problems  in  which  the  given  time  is  near  noon,  great  care  must  be  exercised  in 
determining  the  date  for  which  7P0  is  to  be  taken  from  the  Ephemeris.  In  the  present  case, 
the  astronomical  date  for  the  goth  meridian  is  Oct.  30,  for  the  true  C.  S.  T.  shows  that  the 
instant  of  mean  noon  had  passed;  but  at  Columbia  mean  noon  had  not  yet  arrived.  Since 
R0  is  always  to  be  taken  from  the  Ephemeris  for  the  preceding  local  mean  noon,  the  date  to  be 
used  is  Oct.  29. 

P.M. 

P.M.,  Oct.  30,  civil  date 

Oct.  29,  astronomical 


TR 

Oh 

5m 

17*00 

J7-R 

4 

-  4.82 

C 

.  S.  T. 

0 

5 

21.82 

Z, 

9 

iS.33 

Col. 

M.  S.T. 

23 

56 

3-49 

XL 

H 

27 

40.63 

HIM 

3 

55-91 

fl 

H 

27 

40.03 

OF 

H 

28 

9-75 

J0F 

— 

29.72 

Ans. 

Example  25.  When  the  error  of  a  timepiece,  a,  is  given  and  it  is  required  to  find  the 
corrections  of  two  others  b  and  c,  the  observations  and  reductions  may  be  controlled  by  a 
circular  comparison,  /'.  c.  by  comparing  «  and  b,  a  and  c,  and  b  and  c .  The  first  comparison 
leads  to  the  error  of  b.  The  given  error  of  a,  and  that  calculated  for  b,  may  then  be  used  to 
reduce  the  second  and  third  comparisons.  Each  of  these  leads  to  a  value  of  the  error  of  c  and 
the  two  results  must  agree  within  the  uncertainty  of  the  observations  and  calculations. 

The  Fauth  sidereal  clock,  a  Bond  sidereal  chronometer  and  the  Gregg  and  Rupp  central 
standard  clock  of  the  Laws  Observatory  were  compared  in  this  manner  on  1902,  April  18. 
The  bracketed  numbers  are  the  results  of  the  comparisons.  J#K=  +  im  I4!3,  find  J#B  and 


ioh8m4S!o"\          TG&K  8h  26™  30!S  P.M.  \         TG  *  R  8h   28™  2550     P.M. 

10    7     27.1  J             0v  10       9     35.0                             0K  10     12     47.5 

-f  i     14.3              J0F  i     14.3                           J^B                           -  3.6 

10     8     41.4                0  10     10     49.3                              0  10     12     43.9 

—  3-6         C.  S.  T.  8     35      4.0                      C.  S.  T.  8     36     58.3 

J:TG*R  +8   33.2                 JrG4R          +s   33.3 


CLOCK  COMPARISONS  55 

The  second  and  third  comparisons  are  reduced  by  the  method  used  for  Ex.  23.  The 
details  of  the  conversion  of  0  into  C.  S.  T.  are  omitted.  The  two  values  of  J7"G4R  present 
a  satisfactory  agreement. 

Example  26.  Given  thirty  comparisons  of  a  Waltham  watch  and  a  Bond  sidereal 
chronometer  made  at  intervals  of  one  minute;  to  find  the  rate  per  minute  of  the  watch 
referred  to  the  chronometer,  a  precise  value  of  the  watch  time  corresponding  to  the  first 
chronometer  reading,  and  the  average  uncertainty  of  a  single  comparison. 

The  interval  between  any  two  chronometer  readings  minus'  the  difference  between  the 
corresponding  watch  readings  is  the  loss  of  the  watch  as  compared  with  the  chronometer 
during  the  interval.  The  quotient  of  the  loss  by  the  interval  in  minutes  is  a  value  of  the 
relative  rate  per  minute.  Thus,  if 

7C  =  interval  between  two  chronometer  times, 
7W  =  interval  between  two  watch  times, 
R  =  relative  rate  of  watch  per  minute, 
then 


The  solution  of  the  first  part  of  the  problem  may  therefore  be  accomplished  by  grouping 
the  comparisons  in  pairs  and  applying  equation  (a).  The  mean  of  the  resulting  values  of  R 
will  then  be  the  final  result.  The  selection  of  the  comparisons  for  the  formation  of  the  pairs 
requires  careful  attention  if  the  maximum  of  precision  is  to  be  secured.  To  obtain  a  criter- 
ion for  the  most  advantageous  arrangement,  consider  the  resultant  error  of  observation  in  R 
when  derived  from  equation  («).  Denoting  the  influence  of  the  errors  in  the  observed  watch 
times  upon  the  interval  7W  by  e  we  find  for  the  error  of  R 


Since  e  is  independent  of  the  length  of  the  interval  separating  the  comparisons,  it  follows 
from  (b)  that  the  precision  of  R  increases  with  the  length  of  this  interval. 

It  is  desirable  for  the  sake  of  symmetry  in  the  reduction  that  the  separate  values  of  R 
should  be  of  the  same  degree  of  precision;  and  it  is  important  to  arrange  the  calculation  so  that 
any  irregularity  in  the  relative  rate  will  be  revealed.  The  reduction  will  then  give  not  only 
the  quantitative  value  of  the  final  result,  but  at  the  same  time  will  throw  light  upon  the  reli- 
ability of  the  instruments  employed. 

We  are  thus  led  to  the  following  grouping  of  the  comparisons:  i  and  16,  2  and  17,  3  and 
18,  ......  15  and  30;  or,  in  general,  the  «th  comparison  is  paired  with  the  (15  -j-»)th.     The 

fourth  column  of  the  table  gives  the  values  of  7W  corresponding  to  this  choice.  The  first  of 
these  is  derived  by  subtracting  the  first  T\\  from  the  sixteenth;  the  second,  by  subtracting  the 
second  7*w  from  the  seventeenth,  and  so  on.  The  15  values  of  7\v  substituted  into  equation 
(«),  together  with  the  constant  value  7C  =  I5m,  would  give  15  separate  values  for  R.  The  first 
of  these  would  depend  upon  data  secured  during  the  first  15  minutes  of  the  observing  period; 
the  last,  upon  those  obtained  during  the  last  15  minutes;  while  the  intermediate  values  of  R 
would  correspond  to  various  intermediate  15-minute  intervals.  Any  irregularity  in  the  rate 
will  therefore  reveal  itself  in  the  form  of  a  progressive  change  in  the  separate  values  of  R. 
But,  since  7C  is  assumed  to  be  constant  throughout,  equation  («)  shows  that  constancy  of  7W 
will  be  quite  as  satisfactory  a  test  of  the  reliability  of  the  timepieces  as  constancy  in  R.  It  is 
not  necessary,  therefore,  to  calculate  the  separate  values  of  the  relative  rate;  and  for  the  der- 
ivation of  the  final  result  we  adopt  the  simpler  procedure  of  forming  the  mean  of  the  values 
of  7W,  which  we  then  substitute  into  (a)  with  7C  =  15™.  We  thus  find  mean  7W  =  I4m  57?6.5, 
whence  the  mean  relative  rate  of  the  watch  referred  to  the  chronometer  is  05157  per  minute  of 
chronometer  time. 


56 


PRACTICAL  ASTRONOMY 
WATCH  AND  CHRONOMETER  COMPARISON 


No. 

6K 

TV 

Av 

(n-i)R 

7\v 

V 

i 

oh  45m  ofo 

I0h  25'"   2515. 

i4m  5716 

OfOO 

25590 

—  fil 

2 

46    o.o 

26      25.8 

.6 

o.  16 

25.96 

—•17 

3 

47    O.Q 

27      25.7 

•  4 

0.31 

26.01 

—  .22 

4 

48    o.o 

28      25.5 

•5 

0.47 

25-97 

—.18 

5 

49    o.o 

29      25.1 

.8 

0.63 

25-73 

+  .06 

6 

50    o.o 

30      25.0 

•7 

0.78 

25.78 

+  .01 

7 

51     o.o 

31       24.8 

.8 

0-94 

25-74 

+  -05 

8 

52     o.o 

32      24.7 

•  7 

I.  JO 

25.80 

—  .01 

9 

53    o.o 

33     24.4 

•7 

1.26 

25.66 

+.13 

10 

54    o.o 

34     24-2 

.8 

1.41 

25.61 

+.18 

ii 

55     o.o 

35     24.1 

•7 

i-57 

25.67 

+  .12 

12 

56    o.o 

36     239 

•7 

i-73 

25-63 

+.16 

J3 

57    o.o 

37     23.8 

•7 

1.88 

25.68 

+.ii 

H 

58    o.o 

38     23.7 

•7 

2.04 

25-74 

+-05 

1.5 

59    o-o 

39     23.7 

•4 

2.  2O 

25.90 

—  .11 

16 

I         0      0.0 

40     23.5 

i  5)  9-8 

2.36 

25.86 

—.07 

i7 

I      0.0 

4i"   23-4 

H  57-65 

2.5I 

25-9I 

—  .12 

18 

2       O.O 

42     23.1 

15    o.oo 

2.67 

25-77 

+  .02 

'9 

3    o.o 

43     23.0 

15)  2.35 

2.83 

25-83 

—.04 

20 

4    o.o 

44     22.9 

-ff  =  0*157 

2.98 

25.88 

—.09 

21 

5     o.o 

45     22.7 

3-H 

25.84 

'—  -°5 

22 

6    o.o 

46      22.6 

3-30 

25.90 

—  .11 

23 

7    o.o 

47     22.4 

3-45 

25-85 

—.06 

24 

8    o  o 

48      22.1 

3.61 

25-7I 

+.08 

25 

9    o.o 

49     22.0 

3-77 

25-77 

+  .02 

26 

10    o.o 

50    21.8 

3-92 

25.72 

+  .07 

27 

II      O.O 

51     21.6 

4.08 

25.68 

+  .11 

28 

12      O.O 

52       21.5 

4.24 

25.74 

+  -05 

29 

13    o.o 

53     21.4 

4.40 

25.80 

—  .OI 

30 

14    o.o 

54    21.  i 

4-55 

25-65 

+  .I4 

Precise  "1 
Comp.  / 

o    45    o.oo 

10    25     25.79 

30)23.69 

+  1-36 

—1-35 

M0  = 

25-79 

Rem.  =  —  o.oi           +0.01 

30)  2.71 

Average  Residual  =  =t  0509 

An  examination  of  the  individual  values  of  7W  for  the  given  problem  affords  no  certain 
evidence  of  a  variability  of  the  relative  rate. 

As  for  the  second  requirement  of  the  problem,  it  is  evident  that  were  the  observations  per- 
fectly made,  with  a  watch  whose  relative  rate  was  zero,  the  seconds  and  tenths  of  seconds  of  all 
the  watch  readings  would  have  been  the  same.  Had  they  been  made  with  the  same  errors  of 
observation  as  actually  occurred,  but  with  a  watch  of  zero  relative  rate,  they  would  have  differed 
among  themselves  only  by  the  errors  of  observation.  The  mean  of  all  the  seconds  readings 


CLOCK  COMPARISONS  57 

would  then  have  given  a  precise  value  of  the  watch  time  corresponding  to  the  first  chronom- 
eter reading.  The  given  problem  may  be  reduced  to  this  case  by  correcting  each  watch 
reading  by  the  effect  of  the  rate  during  the  interval  separating  it  from  the  first  observation. ' 
To  accomplish  this  we  have  only  to  add  to  the  readings,  in  order,  the  quantities  o/?,  iR,  2ft, 
....  2<)JR\  or,  in  general,  to  the  wth  reading,  (n  —  i)/f.  The  values  of  these  corrections  are 
in  column  five  of  the  table,  and  the  watch  times,  corrected  for  rate,  in  column  six.  These 
results  are  given  to  two  places  of  decimals  in  order  to  keep  the  errors  ot  calculation  small  as 
compared  with  the  errors  of  observation.  The  mean  of  the  values  of  Tw,  ioh  25™  25579,  is  the 
required  precise  watch  reading  corresponding  to  the  first  chronometer  reading,  oh  45m  Ofoo. 

To  obtain  a  notion  of  the  uncertainty  of  a  single  comparison,  consider  the  corrected  watch 
readings,  7\y  If  the  true  value  of  R  has  been  used  in  applying  the  corrections  for  rate,  and 
if  the  trtie  value  of  the  first  watch  reading  were  known,  the  actual  error  of  this  and  of  each  of 
the  remaining  readings  could  at  once  be  found  by  forming  the  difference  between  the  true 
value  and  each  of  the  corrected  watch  times.  The  average  of  the  errors  would  then  indicate 
the  precision  of  the  comparisons.  But  the  true  values  of  R  and  of  the  first  comparison  are 
not  known  and  cannot  be  found.  We  must  therefore  proceed  as  best  we  may;  and,  accord- 
ingly, we  use  for  the  true  relative  rate  the  value  calculated  above,  and  for  the  true' value  of 
the  first  watch  reading,  the  mean  of  all  the  corrected  readings.  The  differences  between  each 
corrected  watch  time  and  the  mean  of  them  all  are  called  residuals.  The  residuals  will  differ 
but  little  from  the  corresponding  errors,  for  the  calculated  value  of  R  and  the  mean  7\v  will 
differ  but  little  from  the  quantities  they  are  taken  to  represent.  Although  the  average  of  the 
residuals  will  not  exactly  equal  the  average  of  the  errors,  it  may  be  accepted,  nevertheless,  as 
a  measure  of  the  precision  of  the  observations;  for,  barring  a  constant  systematic  error,  it  is 
evident  that  the  more  accurate  the  observations,  i.e.  the  smaller  their  variations  among  them- 
selves, the  less  will  be  the  average  residual. 

Denoting  the  residuals  by  v}  and  the  mean  of  the  corrected  watch  times  by  J/0,  we  have 

v  =  M0-T\v  (0 

The  values  of  v  formed  in  accordance  with  (c)  are  in  the  last  column  of  the  table. 

A  valuable  control  may  be  applied  at  this  point.  It  is  easily  shown  that  if  the  exact  value 
of  M0  be  used  for  the  formation  of  the  residuals,  their  algebraic  sum  must  be  zero.  (Num. 
Comp.  p.  17.)  If,  however,  an  approximation  for  M0  is  used,  the  algebraic  sum  of  the  resid- 
uals will  equal  the  negative  value  of  the  remainder  in  the  division  which  gives  as  quotient 
the  value  used  as  a  mean. 

In  the  present  case  the  algebraic  sum  of  the  residuals  is  -(-  o.oi ;  the  remainder  is  — o.oi, 
which  checks  the  formation  of  the  mean  and  the  residuals.  The  average  residual,  without 
regard  to  algebraic  sign,  is  ±  0509.  This  we  may  accept  as  the  average  uncertainty  of  a  single 
comparison. 

The  principles  illustrated  in  the  preceding  reduction  find  frequent  application  in  the 
treatment  of  the  data  of  observation.  The  example  is  typical  and  the  methods  followed  in  the 
discussion  should  receive  careful  attention.  In  particular,  the  grouping  of  the  observations 
for  the  determination  of  the  mean  value  of  R  should  be  examined;  and  the  student  should 
investigate  for  himself  the  precision  of  the  result  when  such  combinations  of  the  comparisons 
as  i  and  2,  2  and  3,  ....  29  and  30;  i  and  2,  3  and  4,  ....  29  and  30;  i  and  30,  2  and  28,  .... 
15  and  16;  etc,  are  employed  in  place  of  that  actually  used. 

Example  27.  To  determine  the  average  uncertainty  of  a  single  comparison  of  two  time- 
pieces by  the  method  of  coincident  beats. 

Ten  successive  coincidences  of  the  beats  of  a  Bond  sidereal  chronometer  with  those  of  a 
Gregg  &  Rupp  mean  time  clock  are  taken  as  the  basis  of  the  investigation.  The  method  used 
for  the  reduction  is  similar  to  that  employed  in  Ex.  26.  The  comparisons  are  in  the  second 
and  third  columns  of  the  table.  Since  the  chronometer  beats  half-seconds  and  the  clock  sec- 
onds, the  interval  between  the  successive  coincidences  is  that  required  for  the  clock  to  lose 


58 


PRACTICAL  ASTRONOMY 


o?5  as  compared  with  the  chronometer.  Denote  the  true  value  of  this  interval  by  /.  To  ex- 
hibit the  influence  of  the  errors  of  observation  we  find  what  the  clock  readings  would  have 
been  had  they  all  been  made  at  the  same  instant  as  the  first.  This  is  done  by  subtracting 
from  the  readings,  in  order,  o7,  i/,  27,  .  .  . .  gf.  The  numerical  values  of  the  corrections  are 
in  column  five,  and  the  reduced  clock  readings  themselves,  in  column  six.  The  value  to  be 
used  for  7  is  one-fifth  of  the  average  of  the_intervals  between  the  nth  and  the  (w+5)th  clock 
readings.  The  individual  values  of  thes'e  intervals  are  in  column  four.  Their  mean  is 
I4m55?2,  whence  7=  2m59»o4.  The  variations  in  the  values  of  7"  represent  the  influence  of  the 
errors  of  observation.  The  average  residual  tor  the  reduced  clock  readings  is  zfc-  2594,  which 
may  be  accepted  as  the  average  uncertainty  of  the  time  of  a  coincidence.  Since  the  clock 
loses  i6  in  358%  the  corresponding  average  uncertainty  of  a  comparison  is  ±  05008. 

COMPARISON  BY  COINCIDENT  BEATS. 


No. 

6 

r 

'. 

(»-0/ 

7" 

V 

i 

i7h  35m  5750 

2h    ftm    o;o 

i4m     45?o 

Om         050 

2h  5m  6o5o 

-  y-3 

2 

38      52.5 

8    55-° 

62.0 

2          59.0 

56.0 

+  0.7 

3 

4i      53-o 

"    55-o 

60.0 

5       5S.i 

S6  9 

—     0.2 

4 

44      52.5 

14    54-° 

54  -o 

8       57-i 

56-9 

—     0.2 

5 

47      49-0 

17    50.0 

55-0 

ii        56.2 

53-8 

+      2.9 

6 

50      44-5 

20     450 

5)276.0 

M       55-2 

49-8 

+      6.9 

7 

53     57-o 

23    57-o 

5)r4       55-2 

17       54-2 

62.8 

—    6.1 

8 

56      55-5 

26    55.0 

1=     2          59.04 

20       53-3 

61.7 

—    5-o 

9 

59     49  -o 

29    48.0 

23       52-3 

55-7 

+      1.0 

10 

18      2     46.5 

32    45-o 

26       51.4 

536 

±     3-i 

10)567.2           -f  14.6 

^o=      56.7                  -14.8 

Clock  loses  i8  in  358".                                                                                    Rem=:-|-o.2            —    0.2 

\verage  uncertainty  of  a  single                                                                                                  10)    29.4 

:omparison  =  ±  2594/358  —  d=  05008.                                                   Average  Residual  =  ±  2594 

35.  The  care  of  timepieces — All  timepieces  should  be  wound  at  regular  in- 
tervals. They  should  be  protected  from  moisture,  electrical  and  magnetic  in- 
fluences, and  extremes  of  temperature,  especially  the  direct  rays  of  the  sun.  They 
yield  the  best  results  when  at  rest,  absolutely  untouched,  except  as  winding 
may  be  necessary.  Portable  instruments  must  not  be  subjected  to  violent  shocks, 
jolts,  or  oscillatory  motions.  Chronometers  are  particularly  sensitive  to  such  dis- 
turbances, especially  oscillations.  Timepieces  of  this  sort  are  usually  hung  in  gim- 
bals, mounted  in  a  substantial  wooden  case.  When  at  rest,  or  when  subjected  to 
the  long  periodic  motions  of  a  ship,  they  should  hang  free  in  the  gimbals  in  order 
that  the  mechanism  may  remain  constantly  horizontal  in  position.  When  trans- 
ported from  place  to  place  on  land,  the  gimbals  should  be  locked.  Otherwise  the 
unavoidable  jarring  may  produce  oscillations  sufficient  to  change  appreciably  the 
error  and  the  rate.  If  the  journey  is  such  that  shocks  can  not  be  avoided,  it  is 
safer  to  stop  the  instrument  and  insert  thin  wedges  of  cork  between  the  balance 
wheel  and  the  supporting  frame,  using  just  sufficient  force  to  hold  them  in  place. 
In  this  way  the  delicate  pinions  of  the  balance  may  be  guarded  from  injury.  The 


HORIZON  AND   VERNIER  59 

chronometer,  so  far  as  possible,  should  be  kept  in  a  fixed  position  with  respect 
to  the  points  of  the  compass. 

THE  ARTIFICIAL  HORIZON 

36.  Description  and  use. — The  artificial  horizon  consists  of  a  shallow  dish 
filled  with  mercury.    The  force  of  gravity  'brings  the  surface  to  a  horizontal  posi- 
tion, and  the  high  reflective  power  of  the  metal  makes  it  possible  to  see  the  various 
celestial  bodies  reflected  in  the  surface.     Any  given  object  and  its  image  will  be 
situated  on  the  same  vertical  circle,  and  the  angular  distance  of  the  image  below  the 
surface  will  be  equal  to  that  of  the  object  above.    The  angular  distance  between 
the  object  and  its  image  is  therefore  twice  its  apparent  altitude.    Strictly  speaking, 
this  is  true  only  when  the  eye  of  the  observer  is  at  the  surface  of  the  mercury, 
but  for  distant  objects  the  error  is  insensible. 

The  measurement  of  the  distance  between  the  object  and  its  image  therefore 
affords  a  means  of  determining  the  altitude  of  a  celestial  body,  and  in  this  con- 
nection the  artificial  horizon  is  a  valuable  accessory  to  the  sextant.  It  can  also 
be  used  to  advantage  with  the  engineer's  transit  for  the  elimination  of  certain 
instrumental  errors. 

The  artificial  horizon  is  usually  provided  with  a  glass  roof  to  protect  the 
surface  of  the  mercury  from  disturbances  by  air  currents.  It  is  important  that 
the  plates  of  glass  should  be  carefully  selected  in  order  that  the  light  rays  travers- 
ing them  may  not  be  deflected  from  their  course.  The  effect  of  any  non-parallelism 
of  the  surfaces  may  be  eliminated  by  making  an  equal  number  of  settings  with 
the  roof  in  the  direct  and  reversed  position,  reversal  being  accomplished  by  turn- 
ing the  roof  end  for  end. 

THE    VERNIER 

37.  Description  and  theory. — The  vernier  is  a  short  graduated  plate  attached 
to  scales  for  the  purpose  of  reducing  the  uncertainty  of  measurement.     It  takes 
its  name  from  its  inventor,  Pierre  Vernier,  who  in  1631  described  its  construction 
and  use.     In  its  usual  form  the  graduations  are  such  that  the  total  number  of 
vernier  divisions,  which  we  may  denote  by  n,  is  equal  to  n  —  i  divisions  of  the 
scale,  the  graduation  nearest  the  zero  of  the  scale  marking  the  zero  of  the  vernier. 
The  vernier  slides  along  the  scale,  the  arrangement  being,  such  that  the  angle, 
or  length,  to  be  measured  corresponds  to  the  distance  between  the  zeros  of  the 
scale  and  of  the  vernier.    When  the  zero  of  the  vernier  stands  opposite  a  gradua- 
tion of  the  scale,  the  desired  reading  is  given  directly  by  the  scale.    Usually  this 
will  not  occur,  and  the  vernier  is  then  used  to  measure  the  fractional  part  of  the 
scale  division  included  between  the  last  preceding  scale  graduation  and  the  zero 
of  the  vernier. 

The  difference  between  the  values  of  a  scale  and  a  vernier  division  is  called 
the  least  reading  =  /  of  the  vernier.  If 

d  =  value  of  one  division  of  the  scale, 
d'  =  value  of  one  division  of  the  vernier, 


60  PRACTICAL  ASTRONOMY 

then,  for  the  method  of  graduation  described  above, 

(n  —  I  )  d  =  nd' 
whence 


The  least  reading  of  the  vernier  is  therefore  i/nth  of  the  value  of  a  scale  division. 
Now,  for  an  arbitrary  setting  of  the  vernier,  consider  the  intervals  between 
the  various  vernier  graduations  and  the  nearest  preceding  graduations  of  the  scale, 
beginning  with  the  zero  of  the  vernier  and  proceeding  in  order  in  the  direction 
of  increasing  readings.  The  first  interval  is  the  one  whose  magnitude  is  to  be 
determined  by  the  vernier.  Denote  its  value  by  "<.'.  Since  a  vernier  division  is 
less  than  a  scale  division  by  th'e  least  reading,  /,  it  follows  that  the  interval  between 
the  second  pair  of  graduations  will  be  v  —  /;  that  between  the  third  v  —  2/;  and  so 
on,  each  successive  interval  decreasing  by  /.  By  proceeding  far  enough  we  shall 
find  a  pair  for  which  the  interval  differs  from  zero  by  an  amount  equal  to,  or 
less  than  //2,  a  quantity  so  small  that  the  graduations  will  nearly,  if  not  quite, 
coincide.  Suppose  this  pair  to  be  n'  divisions  from  the  zero  of  the  vernier.  The 
value  of  the  corresponding  interval  will  be  v  —  ;*'/—£,  and  we  therefore  find 


v  = 
In  practice  we  disregard  s  and  use 

v  =  n'l.  (70) 

To  determine  the  value  of  v,  therefore,  we  count  the  number  of  vernier 
divisions  from  the  zero  of  the  vernier  to  the  vernier  graduation  which  most  nearly 
coincides  with  a  graduation  of  the  scale.  The  product  of  this  number  into  the 
least  reading  is  the  value  of  v.  The  final  result  is  the  sum  of  v  and  the  reading 
corresponding  to  the  last  scale  graduation  preceding  the  zero  of  the  vernier. 

In  practice  the  actual  counting  of  the  number  of  divisions  between  the  zero 
of  the  vernier  and  the  coincident  pair  is  avoided  by  making  use  of  the  numbers 
stamped  on  the  vernier.  These  give  directly  the  values  of  n'l  corresponding  to 
certain  equidistant  divisions  of  the  vernier.  Usually  one  or  two  divisions  precede 
the  zero  and  follow  the  last  numbered  graduation  of  the  vernier.  These  do  not 
form  a  part  of  the  n  divisions  of  the  vernier,  and  are  therefore  to  be  disregarded 
in  the  determination  of  /.  They  are  added  to  assist  in  the  selection  of  the  coin- 
cident pair  when  coincidence  occurs  near  the  end  of  the  vernier. 

38.  Uncertainty  of  the  result. — The  error  of  a  reading  made  with  a  per- 
fectly constructed  vernier  is  s,  whose  maximum  absolute  value  is  1/2.  The  uncer- 
tainty of  the  result  is  therefore  1/2. 

The  gain  in  precision  resulting  from  the  use  of  the  vernier  may  be  found 
by  comparing  the  uncertainty  of  its  readings  with  that  arising  when  the  scale 
alone  is  used.  The  latter  may  be  fixed  at  o.O5d,  as  experience  shows  that  this 


UNCERTAINTY  OF  VERNIER  READINGS  61 

is  approximately  the  uncertainty  of  a  careful  eye  estimate  of  the  magnitude  of  r. 
The  inverse  ratio  of  the  two  uncertainties  may  be  taken  as  a  measure  of  the 
increase  in  precision,  whence  we  find  that  the  result  given  by  the  vernier  is  approx- 
imately ;//io  times  as  precise  as  that  derived  from  an  estimate  of  the  fractional 
parts  of  a  scale  division.  It  appears,  therefore,  that  a  vernier  is  of  no  advantage 
unless  the  number  of  its  divisions  is  in  excess  of  ten. 

The  use  of  a  magnifying  lens  usually  shows  that  none  of  the  vernier  grad- 
uations exactly  coincides  with  a  graduation  of  the  scale.  With  a  carefully  grad- 
uated instrument,  it  is  possible,  by  estimating  the  magnitude  of  e,  to  push  the 
precision  somewhat  beyond  the  limit  given  above.  To  do  so  it  is  only  necessary 
to  compare  s  with  the  interval  between  the  next  following  pair  of  graduations, 
or  with  that  of  the  pair  immediately  preceding,  according  as  e  is  positive  or 
negative.  The  sum  of  the  two  intervals  to  be  compared  is  /.  It  is  therefore 
possible  to  estimate  £  in  fractional  parts  of  the  least  reading. 

The  condition  that  n  divisions  of  the  vernier  equal  n — I  divisions  of  the 
scale  must  be  rigorously  fulfilled  if  reliable  results  are  to  be  obtained.  The  matter 
should  be  tested  for  different  parts  of  the  scale  by  bringing  the  zero  of  the  vernier 
into  coincidence  with  a  scale  graduation,  and  then  examining  whether  the  (//-(-i)st 
vernier  graduation  stands  exactly  opposite  graduation  of  the  scale.  Information 
may  thus  be  obtained  as  to  the  accuracy  with  which  the  graduation  of  the  instru- 
ment has  been  performed. 

The  vernier  should  lie,  preferably,  in  the  same  plane  as  the  scale,  and,  in 
all  positions,  should  fit  snugly  against  the  latter.  In  many  instruments,  however, 
it  rests  on  top,  the  plate  being  beveled  to  a  knife  edge  where  it  touches  the  scale. 
With  this  arrangement  the  greatest  care  must  be  exercised  in  reading  to  keep 
the  line  of  sight  perpendicular  to  the  scale.  Otherwise  an  error  due  to  parallax 
will  affect  the  result. 

THE  ENGINEER'S  TRANSIT 

39.  Historical. — The  combination  of  a  horizontal  circle  with  a  vertical  arc 
for  the  measurement  of  azimuth  and  altitude  is  known  to  have  been  used  by  the 
Persian  astronomers  at  Meraga  in  the  thirteenth  century,  and  it  is  possible  that 
a  similar  contrivance  was  employed  by  the  Arabs  at  an  even  earlier  date.  The 
principle  involved  did  not  appear  in  western  Europe,  however,  until  the  latter 
half  of  the  sixteenth  century.  There  it  found  its  first  extensive  application  in  the 
instruments  of  Tycho  Brahe,  who  constructed  a  number  of  "azimuth-quadrants" 
for  his  famous  observatory  on  the  island  of  Hveen.  The  vertical  arcs  of  Tycho's 
instruments  were  movable  about  the  axis  of  the  horizontal  circle,  and  were  pro- 
vided with  index  arms  fitted  with  sights  for  making  the  pointings.  The  adjust- 
ment for  level  was  accomplished  by  means  of  a  plumb  line,  the  spirit  level  not  yet 
having  been  invented.  Magnification  of  the  object  was  impossible,  as  a  quarter 
of  a  century  was  still  to  elapse  before  the  construction  of  the  first  telescope.  The 
instruments  were  large  and  necessarily  fixed  in  position;  and,  indeed,  there  was 
no  need  for  moving  them  from  place  to  place  as  they  were  intended  solely  for 
astronomical  observations.  Though  primitive  in  design,  they  were  constructed 


62  PRACTICAL  ASTRONOMY 

with  the  greatest  care,  and  were  capable  of  determining  angular  distances  with  an 
uncertainty  of  only  i'  or  2'.  They  are  of  interest  not  only  on  account  of  the  re- 
markable series  of  results  they  yielded  in  the  hands  of  Tycho,  but  also  because 
they  embody  the  essential  principle  of  the  modern  altazimuth,  the  universal  instru- 
ment, the  theodolite,  the  engineer's  transit,  and  a  variety  of  other  instruments. 

None  of  these  modern  instruments  is  the  invention  of  any  single  person, 
but  rather  a  combination  of  inventions  by  various  individuals  at  different  times.  The 
telescope,  first  constructed  during  the  early  years  of  the  seventeenth  century,  was 
adapted  to  sighting  purposes  through  the  introduction  of  the  reticle  by  Gascoigne, 
Auzout,  and  Picard.  Slow  motions  were  introduced  by  Hevelius.  The  vernier  was 
invented  in  1631,  and  the  spirit  level,  by  Thevenot,  in  1660.  All  these  were  com- 
bined with  the  principle  of  the  early  azimuth-quadrant  to  form  the  altazimuth, 
which  appears  first  to  have  been  made  in  a  portable  form  by  John  Sisson,  an 
Englishman,  about  the  middle  of  the  eighteenth  century.  At  the  beginning  of  the 
nineteenth  century  the  design  and  construction  were  greatly  improved  by  Reich- 
enbach,  who  also  added  the  movable  horizontal  circle,  thus  making  it  possible 
to  measure  angles  by  the  method  of  repetitions.  The  universal  instrument  was 
then  practically  complete,  and  the  transition  to  the  engineer's  transit  required 
only  the  addition  of  the  compass  and  such  minor  modification  as  would  meet 
the  requirements  of  precision  and  portability  fixed  by  modern  engineering 
practice. 

For  a  detailed  description  of  the  engineer's  transit,  the  student  is  referred 
to  any  standard  work  on  surveying.  Certain  attachments,  notably  the  compass 
and  the  telescope  level,  are  not  required  for  the  determination  of  latitude,  time, 
and  azimuth.  On  the  other  hand,  it  is  desirable  that  the  instrument  used  in  the 
solution  of  these  problems  should  possess  features  not  always  present  in  the  mod- 
ern instrument.  In  particular,  the  vertical  circle  should  be  complete,  and  should 
be  provided  with  two  verniers  situated  180°  apart.  A  diagonal  prism  for  the 
observation  of  objects  near  the  zenith,  and  shade  glasses  for  use  in  solar  obser- 
vations are  a  convenience,  though  not  an  absolute  necessity. 

40.  Influence  of  imperfections  of  construction  and  adjustment. — It  is  assumed 
that  the  student  is  familiar  with  the  methods  by  which  the  engineer's  transit  may 
be  adjusted,  and  that  observations  will  not  be  undertaken  until  the  various  adjust- 
ments have,  been  made  with  all  possible  care.  But  since  an  instrument  is  never 
perfect,  it  becomes  of  importance  to  determine  the  influence  of  the  residual  errors 
in  construction  and  adjustment,  and  to  establish  precepts  for  the  arrangement  of 
the  observing  program  such  that  this  influence  may  be  reduced  to  a  minimum. 

In  the  instrument  fulfilling  the  ideal  of  construction  and  adjustment,  the  fol- 
lowing conditions,  among  others,  are  satisfied  : 

1.  The  rotation  axes  of  the  horizontal  circle  and  the  alidade  coincide. 

2.  The  planes  of  the  circles  are  perpendicular  to  the  corresponding  axes 
of  rotation. 

3.  The  centers  of  the  circles  lie  in  the  corresponding  axes  of  rotation,  and 

the  lines  joining  the  zeros  of  the  verniers  pass  through  the  axes. 


INSTRUMENTAL  ERRORS 


63 


4- 

5- 
6. 


7- 


The  vertical  axis  of  rotation  is  truly  vertical  when  the  plate  bubbles  are 
centered. 

The  horizontal  rotation  axis  is  perpendicular  to  the  vertical  axis. 
The  line  of  sight,  i.e.  the  line  through  the  optical  center  of  the  objective 
and  the  middle  intersection  of  the  threads,  is  perpendicular  to  the  hori- 
zontal axis. 

The  vertical  circle  reads  zero  when  the  line  of  sight  is  horizontal. 
It  is  the  task  of  the  instrument  maker  to  see  that  the  first  three  of  these  con- 
ditions are  satisfied.     The  observer,  on  the  other  hand,  is  responsible  for  the  re- 
mainder. 

No.  i  is  of  importance  only  in  the  measurement  of  horizontal  angles  by  the 
method  of  repetitions.  The  error  arising  in  such  measures  from  non-coincidence 
of  the  vertical  axes  may  be  eliminated  by  the  arrangement  of  the  observing  pro- 
gram described  in  Section  47. 

No.  2.  It  can  be  shown  that  the  error  due  to  lack  of  perpendicularity  of  the 
circles  to  the  axes  is  of  the  order  of  the  square  of  the  deviation.  In  well  con- 
structed instruments  it  is  therefore  insensible. 

No.  3.  If  the  third  condition  is  not  satisfied  the  readings  will  be  affected  by 
an  error  called  eccentricity. 


Fig.  7. 

In  Fig.  7  let  C  be  the  center  of  the  graduated  circle  OV]_V2  !  <?>  the  point 
where  the  rotation  axis  intersects  the  plane  of  the  circle;  O,  the  zero  of  the 
graduations;  and  V^  and  F2  the  zeros  of  the  verniers.  The  distance  aC  =  e 
is  the  eccentricity  of  the  circle.  The  perpendicular  distance  of  a  from  the 
line  joining  Vl  and  V .,  is  the  eccentricity  of  the  verniers.  The  reading  of  l/l 
is  the  angle  OCF:,  and  of  F2,  OCF2.  Denote  these  by  R^  and  R2,  respect- 
ively. The  angles  through  which  the  instrument  must  be  rotated  in  order 
that  the  zeros  of  the  verniers  may  move  from  O  to  the  positions  indicated,  are 

l^=Al  and  OaV '.2=A.2,  respectively.    At  and  A2  are  therefore  to  be  regarded 


64  PRACTICAL  ASTRONOMY 

as  the  angles  which  determine  the  positions  of  the  verniers  with  respect  to  O 
for  the  pointing  in  question.  The  relations  connecting  A1  and  A,,  with  the  vernier 
readings,  R:  and  R2,  are 


where  E0,  E1}  and  E2  are  the  corrections  for  eccentricity  for  the  points  O, 
and  Vz.    The  mean  of  (71)  and  (72)  is 


R,)  +  E.  +  ^(E,-EI).  (73) 

For  any  other  pointing  of  the  telescope,  we  have  the  analogous  equation 

+  £0  +  y2(E2'-EI').       (74) 


It  is  easily  shown  that  E2  —  El  and  E.,'  —  £/  are  of  the  order  of  ee'/r2  ,  where 
e'  is  the  eccentricity  of  the  verniers  and  r  the  radius  of  the  circle.  The  last  terms 
of  (73)  and  (74)  are  entirely  insensible  in  a  well  constructed  instrument.  The 
difference  of  (73)  and  (74)  is  therefore 

#(A,'+A,')  -  %(At+A.)  =  #(*/+*,')  -  X(Rt+R,).  (75) 

The  left  member  of  (75)  is  the  angular  distance  through  which  the  instru- 
ment is  rotated  in  passing  from  the  first  position  to  the  second,  and  the  equation 
shows  that  this  angle  is  equal  to  the  difference  in  the  means  of  the  vernier  read- 
ings for  the  final  and  initial  positions.  The  eccentricity  is  therefore  eliminated 
by  combining  the  means  of  the  readings  of  both  verniers. 

It  can  be  shown  that  the  eccentricity  will  also  be  eliminated  by  combining 
the  means  of  any  number  of  verniers,  greater  than  two,  uniformly  distributed 
about  the  circle.  In  practice  it  is  sufficient  to  use  the  degrees  indicated  by  the 
first  vernier  with  the  means  of  the  minutes  and  seconds  of  the  two  readings. 

Nos.  4  —  7.  Horizontal  Angles:  In  the  measurement  of  horizontal  angles 
an  error  of  adjustment  in  No.  7  has  no  influence.  To  investigate  the  effect  of 
residual  errors  in  Nos.  4  —  6,  let 

i=  inclination  of  the  vertical  axis  to  the  true  vertical, 
90°  —  /^inclination  of  the  horizontal   axis   to   the   vertical   axis, 

&r=inclination  of  the  horizontal  axis  to  the  horizon  plane, 
90°  +  ^^inclination  of  the  line  of  sight  to  the  horizontal  axis. 

The  quantities  b  and  c  are  the  errors  in  level  and  collimation,  respectively. 
Then,  in  Fig.  8,  which  represents  a  projection  of  the  celestial  sphere  on  the  plane 
of  the  horizon,  let  Z  be  the  zenith,  Z'  the  intersection  with  the  celestial  sphere 
of  the  vertical  axis  produced,  O  an  object  whose  zenith  distance  is  ZQ,  and  A 
the  intersection  of  the  horizontal  axis  produced  with  the  celestial  sphere  when 
O  is  seen  at  the  intersection  of  the  threads.  The  sides  of  the  triangles  ZAZ' 


UNIVERSITY 

Of 


INSTRUMENTAL  ERRORS 


65 


and  ZAO  have  the  values  indicated  in  the  figure.     Finally,  let  k,  K  and  /  be  the 
directions  of  ZA,  ZO,  and  Z'A,  respectively,  referred  to  ZP. 
Applying  equations  (13)  and  (15)  to  triangle  ZAZ',  we  find 


sin  b  =  sin/  cos  i  -{-  cosy' sin  /cos  /, 
cos  b  sin  k  =  cosy  sin  /. 


(76) 
(77) 


In  a  carefully  adjusted  instrument  i,  j,  and  b  are  very  small,  and  we  may  neglect 
their  squares  as  insensible.     Equations  (76)  and  (77)  thus  reduce  to 


b  =j  -(-/cos/, 


(78) 
(79) 


Equation  (13)  applied  to  triangle  ZAO  gives 

—  sin  c  =  sin  b  cos  £0  -f-  cos  b  sin  z0  cos  (K —  k).  (80) 

Since  c  and  90° — K -{- k  are  also  very  small,  equation  (80)  may  be  written 
—  c  =  b  cos  s0  -f-  (90°  —  K  +  k)  sin  ^0 


or 


—  90° 


=  b  cot  za-\-  c  cosec  z0. 


(81) 


Were  there  no  errors  of  adjustment,  the  direction  of  A  referred  to  P 
would  be  AT — 90°.  The  direction  given  by  the  instrument,  determined  by  the 
angle  through  which  it  must  be  rotated  to  bring  A  from  coincidence  with  ZP 
to  its  actual  position,  is  /.  Since  the  verniers  maintain  a  fixed  position  with  re- 
spect to  A,  the  difference  K — 90° — /  represents"  the  effect  of  the  residual  errors 


66  PRACTICAL  ASTRONOMY 

on  the  horizontal  circle  readings.  But  by  (79)  l=k,  sensibly,  whence  it  follows 
that  the  amount  of  the  error  is  given  by  (81).  If,  therefore,  R  be  the  actual 
horizontal  circle  reading,  and  R^,  the  value  for  a  perfectly  adjusted  instrument. 
we  have 

Ro  =  R  +  ffcot  z0  +  c  cosec  *0,  C.  R.         (82) 

in  which  the  value  of  b  is  given  by  equation  (78).  Assuming  that  equation 
(82)  refers  to  that  position  of  the  instrument  for  which  the  vertical  circle  is  on 
the  right  as  the  observer  stands  facing  the  eyepiece  (C.  R.),  we  find  by  a 
precisely  similar  investigation  for  circle  left  (C.  L.), 


i  =    — 
R0  =  Rl  —  b1cotz0  —  c  cosec  za,         C.  L. 

where  Rl  is  the  circle  reading  less  180°,  and  b^,  the  inclination  of  the  horizontal 
axis  to  the  plane  of  the  horizon  for  C.  L.  The  mean  of  equations  (82)  and 
(84)  is 


0,  (85) 

or,  substituting  the  values  of  b  and  b1  from  (78)  and  (83) 

(86) 


It  therefore  appears  that  the  mean  of  the  readings  of  the  horizontal  circle 
taken  C.  R.  and  C.  L.  for  settings  on  any  object  is  free  from  the  influence  of 
j,  c,  and  the  component  of  i  in  the  direction  of  the  line  of  sight,  viz.,  i  sin  /..  More- 
over, for  objects  near  the  horizon  the  effect  of  /cos/,  the  component  of  i  par- 
allel to  the  horizontal  axis,  is  small,  for  it  appears  in  (86)  multiplied  by  cot  ^ft. 

If  the  instrument  be  provided  with  a  striding  level,  the  values  of  b  and  /?, 
may  be  determined  by  observation.  Their  substitution  into  (85)  will  then  give 
the  horizontal  circle  reading  completely  freed  from  i,  j,  and  c- 

The  readings  may  also  be  freed  from  the  influence  of  b  by  combining  the 
results  of  a  setting  on  O  with  those  obtained  by  pointing  on  the  image  of  O  seen 
reflected  in  a  dish  of  mercury,  both  observations  being  made  in  the  same  position 
of  the  instrument,  either  C.  R.  or  C.  L.  The  reflected  image,  0',  will  be  on  the 
vertical  circle  through  O,  and  as  far  below  the  horizon  as  O  is  above.  Since  the 
horizontal  axis  is  not  truly  horizontal,  it  will  be  necessary  to  rotate  the  instru- 
ment slightly  about  the  vertical  axis  in  turning  from  O  down  to  0  '.  A  will  thus 
move  a  small  amount  to  a  new  position  A'. 

To  investigate  the  effect  of  the  errors  for  a  pointing  on  O'  we  must  therefore 
consider  the  triangle  A'ZO'  in  place  of  AZO  in  Fig.  8.  The  sides  of  A'ZO'  are 
ZA'  =  ZA  =  gtf—  b,  A'0'  =  AO  =  9Q°+c,  and  ZO'  =  180°—  SQ.  The  angle  at 
Z  is  K  —  k'  where  k'  is  the  direction  of  ZA'  referred  to  ZP.  We  then  find,  simi- 
larly to  equation  (80), 


INSTRUMENTAL  ERRORS  67 

—  sin  c  =  —  sin  $cos  z0  -j-  cos  £  sin  z0cos  (A" — /£'), 
whence 

A' —  90°  —  k'  —  —  b  cot  s0  -\-  c  cosec  £„, 

and,  finally,  if  /?'  be  the  horizontal  circle  reading  for  the  setting  on  O' 

R0=R'—  b  cot  z0  +  c  cosec  z0.          C.  R.         (87) 
Equations  (82)  and  (87)  both  refer  to  C.  R.    Their  mean  is 

Ro  =  %  (R  +  /?')  +  c  cosec  s0,          C.  R.        (88) 

By  the  same  method  we  find   from  the  reflected  observation,  C.   L., 
R0  =  A?/  -f-  ^  cot  z0  —  c  cosec  z0,          C.  L.         (89) 

in  which  Rt'  is  the  circle  reading  less  180°   for  C.  L.     This  equation  combined 
with   (84)   gives 

Ro  =  y,(Rt  +  Rt')  —  c  cosec  z0,          C.  L.         (90) 

Equations  (88)  and  (90)  show  that  the  mean  of  the  horizontal  circle  read- 
ings for  direct  and  reflected  observations  of  an  object  in  the  same  position  of 
the  instrument  is  free  from  the  influence  of  any  adjustment  error  in  level. 

Finally  the  combination  of   (88)    and    (90)   gives 

R0=y4(R  +  R'+R>-\-R:\  (91) 

in  other  words  the  mean  of  the  readings,  direct  and  reflected,  for  both  positions 
of  the  instrument,  is  free  not  only  from  b,  but  from  the  collimation  error  as  well. 
Vertical  Circle  Readings:  To  investigate  the  influence  of  /,  /  and  c  upon 
the  readings  of  the  vertical  circle,  consider  again  Fig.  8.  The  true  zenith  dis- 
tance of  0  is  Z0=zt};  that  given  by  the  vertical  circle  readings  is  equal  to  the 
angle  Z'AO.  From  the  triangle  ZAO  we  find 

cos  ~0  —  —  sin  $sin  c  -\-  cos  b  cos  c  cos  (ZAO}. 

The  squares  and  products  of  the  errors  of  adjustment  are  ordinarily  quite  insen- 
sible, whence  we  find  with  all  necessary  precision. 

^or=  Angle  ZAO. 

Denoting  the   instrumental   zenith   distance  Z'AO   by  r,  we  find  £0  —  £=  angle 
ZAZ ',  and  from  triangle  ZAZ' 

cos  $sin  (za  —  s]  —  sin  i sin  /, 


G8  PRACTICAL  ASTROXOMY 

or,  since  b,  z0  —  #,  and  i  are  very  small, 

z0  =  s  +  i  sin  /,  C.  R.         (92) 

A  similar  investigation  gives  for  the  reversed  position  of  the  instrument 

*„  =  *,  +  , 'sin/,  C.L.         (93) 

in  which  z^  is  the  instrumental  zenith  distance  for  C.  L. 

The  angles  z  and  zv  are  not  read  directly  from  the  circles.  The  ordinary 
engineer's  transit  reads  altitudes,  but  if  there  is  any  deviation  from  the  condi- 
tion expressed  in  No.  7,  the  readings  will  not  be  the  true  altitudes,  for  they  will 
include  the  effect  of  the  index  error.  If  r  and  rl  be  the  vertical  circle  readings 
for  C.  R.  and  C.  L.,  respectively,  and  /  the  reading1  when  the  line  of  sight  is  hor- 
izontal, we  have 

s  =go°—r  +/,  C.  R.         (94) 

£,=90°  —  rt  —  /,  C.  L.         (95) 

Substituting  (94)   and  (95)   into   (92)   and   (93) 

so  =  9o°  —  r  +  /+  /sin  /,  C.  R.         (96) 

z0  —  90°  —  rt  —  /  +  /  si  n  /,  C.  L.         (97) 

The  mean  of  (96)  and  (97)  is 

z0  =  90°  —  */z (r  +  rt)  +  i  sin  /.  (98) 

For  an  instrument  whose  vertical  circle  is  graduated  continuously  from  o° 
to  360°  it  is  easily  shown  that  the  equation  corresponding  to  (98)  is 

*«,  =  ^  (^.  —  7',-)  +  i  si  n  /,  (99) 

in  which  z\  and  v2  are  the  circle  readings,  the  subscripts  being  assigned  so  that 


It  therefore  appears  that  the  vertical  circle  readings  are  not  sensibly 
affected  by  /,  c,  or  the  component  of  i  parallel  to  the  horizontal  axis.  The  com- 
ponent of  i  in  the  direction  of  the  line  of  sight,  viz.,  /sin/  enters  with  i'ts  full 
value,  and  (98)  and  (99)  show  that  it  cannot  be  eliminated  even  when  readings 
taken  C.  R.  and  C.  L.  are  combined.  The  formation  of  the  mean  for  the  two 
positions  of  the  instrument  does  eliminate  the  index  error,  however,  i.e.  the 
residual  error  of  adjustment  in  No.  7. 

To  free  the  results  from  z'sin/  we  may  combine  observations  direct  and 
reflected,  using  the  mercurial  horizon.  Considering  the  triangle  A'ZO'  previously 
defined,  we  find  for  the  reflected  observation 

cos  (180°  —  £0)  =  —  sin^sin^  +  cos  b  cos  c  cos  (ZA'O'} 


INSTRUMENTAL  ERRORS  69 

whence,  neglecting  products  and  squares  of  the  errors  of  adjustment,  the  true 
zenith  distance  of  0'  is  180° — z0=  angle  ZA' 0'.  Denoting  the  instrumental 
zenith  distance  of  O' ,  which  is  the  angle  Z'A'O',  by  180°—  s'  we  find 

Angle  ZA'Z'  =  (i8o°  —  z0}  —  (180°  —  z')  =  z'  —  z0. 

In    the    triangle  ZA'Z'  the   sides    are  ZA' =  90°  —  b,  Z'A'-—go° — _/,  and 
ZZ'  =  t,  and  denoting  the  angle  ZZ'A'  by  /'we  find 

cos  $  sin  (#'  —  '£„)  =  sin  /sin  /', 
or  with  sufficient  approximation 

z0  —  z'  —  z'sin/',  C..R.         (100) 

Now  if  r'  be  the  vertical  circle  reading  for  C.  R.,  reflected,  and  /  the  cir- 
cle reading  when  the  line  of  sight  is  horizontal,  we  shall  have,  similarly  to  (94), 


z'  ==  90°  —  r'  —  I,  C.  R. 

This  substituted  into  (100)  gives 

£0  =  900  —  r'  —  / — /sin/'  C.  R.         (102) 

Since  /'  differs  from  /  by  a  quantity  of  the  order  of  the  errors,  the  difference 
between  i  sin  /  and  i  sin  /'  will  be  insensible,  so  that  when  equations  (96)  and 
(102)  are  combined  to  form  the  mean  we  have  simply 

*,  =  90°  —  %  (r  +  r').  C.  R.         ( 103) 

Similar    considerations    for    observations    direct   and    reflected,    C.    L.,    give 

«o  r=  90°  —  lA(rI  +  r/).  C.  L.         (104) 

In  other  words,  the  formation  of  the  means  of  the  vertical  circle  readings  for 
observations  direct  and  reflected  in  the  same  position  of  the  instrument  elimi- 
nates not  only  the  component  of  i  in  the  direction  of  the  line  of  sight,  but  the 
index  correction  as  well.  The  influence  of  z'cos/,  j  and  c  is  insensible.  So 
far  as  the  errors  here  considered  are  concerned,  observations  direct  and  reflected 
in  a  single  position  of  the  instrument  are  sufficient.  Nevertheless  it  is  desirable 
that  measures  be  made  both  C.  R.  and  C.  L.  for  in  this  way  different  parts  of 
the  vertical  circle  are  used,  thus  partially  neutralizing  errors  of  graduation. 

For  an  instrument  with  a  vertical  circle  graduated  continuously  from  o°  to 
360°,  it  is  easily  shown  as  before  that  in  (103)  and  (104)  the  sum  of  the  circle 
readings  must  be  replaced  by  their  difference  taken  in  such  a  way  that  it  is  less 
than  180°. 


70  PRACTICAL  ASTRONOMY 

The  preceding  discussion  assumes  that  the  adjustments  of  the  instrument 
remain  unchanged  throughout  the  observations.  If  this  is  not  so,  the  elimination 
of  the  errors  will,  in  general,  be  incomplete. 

It  is  not  always  convenient  to  make  use  of  the  artificial  horizon,  and  it  is 
therefore  desirable  to  be  able  to  apply  a  method  of  elimination  which  does  not 
depend  upon  this  accessory.  •* 

It  is  easily  shown  that  if  the  instrument  be  rclerelled  before  observing  in 
the  reversed  position,  the  mean  of  the  readings  C.  R.  and  C.  L.,  both  for  the 
horizontal  and  the  vertical  circle,  will  be  free  from  the  errors  in  all  of  the  ad- 
justment under  Nos.  4 — 7,  within  quantities  of  the  order  of  the  products  and 
squares  of  the  errors.  The  same  will  be  true,  even  though  the  plate  bubbles 
are  not  accurately  centered  during  the  direct  observations,  provided,  after  re- 
versal, they  be  brought  to  the  same  position  in  the  tubes  that  they  occupied 
before. 

That  such  will  be  the  case  follows  from  a  consideration  of  Fig.  8.  The 
reversal  and  relevelling  is  equivalent  to  rotating  the  triangle  ZAZ'  about  Zthrough 
the  angle  i8o°-|-2r,  its  dimensions  remaining  unchanged.  A  thus  assumes  a 
new  position  Al}  distant  from  O  by  90°-)-^,  and  Z'  a  position  Z/.  The  triangle 
ZA^O  leads  to  an  equation  differing  from  (84)  only  in  that  bi  is  replaced  by  b. 
The  mean  of  the  new  equation  and  (82)  is  simply 

R.=  X(R  +  R^  (105) 

where  R  and  Rl  are  the  horizontal  circle  readings,  the  latter  having  been  reduced 
by  1 80°.  The  result  is  therefore  free  from  both  b  and  c. 

Again,  from  triangle  ZA^Z^,  we  find  for  circle  left  analogously  to  (97), 

z0  =  C)<f — rt — / — /sin/,  C.  L.         (106) 

in  which  the  vertical  circle  reading  rl  is  not  the  same  as  the  r1  of  (97),  for 
(106)  presupposes  that  the  instrument  is  relevelled  after  reversal,  while  (97) 
assumes  that  no  change  is  made  in  the  position  of  the  vertical  axis  during  the 
observations.  The  mean  of  (96)  and  (106)  is 

•So  =  90  —  %(r  +  r^  (107) 

which  is  free  from  b,  c,  and  /.  For  a  circle  graduated  continuously  we  have 
similarly, 

s0=y*(Vi  —  v^  (i  08) 

where  as  before  the  readings  are  to  be  taken  in  such  an  order  that  their  dif- 
ference is  less  than  180°. 

It  is  assumed  throughout  that  the  pointings  are  always  made  by  bringing 
the  object  accurately  to  the  intersection  of  the  threads.  It  is  important  that  this 
be  done,  even  though  the  threads  be  respectively  horizontal  and  vertical;  for 
observing  at  one  side  of  the  field  is  equivalent  to  introducing  an  abnormal  value 


INSTRUMENTAL  ERRORS— THE  LEVEL  71 

of  the  collimation,  while  pointings  above  or  below  the  horizontal  thread  corre- 
spond to  a  modification  of  the  index  error  of  the  vertical  circle. 

41.  Summary  of  the  preceding  section. — The  preceding  results  may  be 
summarized  as  follows: 

No.  i.  Non-coincidence  of  vertical  axes  enters  only  when  the  horizontal 
circle  is  used  by  the  method  of  repetitions.  Error  eliminated  by  proper  arrange- 
ment of  observing  program.  See  Section  47. 

No.  2.  Non-perpendicularity  of  circles  to  axes  usually  has  no  sensible 
influence  on  circle  readings. 

No.  3.  Eccentricity  of  circles  and  verniers  eliminated  by  forming  means 
of  readings  of  both  verniers.  See  equation  (75). 

Nos.  4 — 7.  Horizontal  circle  readings:  Component  of  deviation  of  vertical 
axis  from  vertical  in  direction  of  line  of  sight,  non-perpendicularity  of  axes,  and 
collimation  eliminated  by  forming  mean  of  readings  taken  C.  R.  and  C.  L.  Com- 
ponent of  deviation  from  vertical  which  is  parallel  to  horizontal  axis  appears  mul- 
tiplied by  cot^0.  See  equation  (86).  Correction  for  the  latter  may  be  made  by 
observations  with  the  striding  level.  See  equation  (85).  All  errors  in  Nos.  4 — 6 
eliminated  by  forming  mean  of  readings  direct  and  reflected,  for  both  C.  R.  and 
C.  L.  See  Equation  (91).  The  error  in  No.  7 — index  error  of  vertical  circle- 
does  not  enter.  Vertical  circle  readings:  All  errors  in  Nos.  4 — 7  excepting  com- 
ponent of  deviation  of  vertical  axis  from  vertical  in  direction  of  line  of  sight 
insensible  or  eliminated  from  mean  of  readings  C.  R.  and  C.  L.  See  equation 
(98)  or  (99).  All  errors  in  Nos.  4 — 7  insensible  or  eliminated  from  mean  of 
readings,  direct  and  reflected,  in  same  position  of  instrument.  See  equations 
(103)  and  (104).  Desirable  to  observe  both  C.  R.  and  C.  L.,  however,  to  reduce 
graduation  error  of  vertical  circle. 

All  errors  under  Nos.  4 — 7  eliminated  from  mean  of  readings  C.  R.  and 
C.  L.  for  both  horizontal  and  vertical  angles  provided  plate  bubbles  have  same 
position  in  tubes  for  both  positions  of  the  instrument.  See  equations  (105) 
and  (107)  or  (108). 

42'  The  level. — The  adjustment  of  the  engineer's  transit  with  respect  to 
the  vertical  is  usually  made  by  means  of  the  plate  bubbles,  any  residual  error 
being  eliminated  by  some  one  of  the  methods  of  Section  40.  In  some  cases, 
however,  it  is  desirable  to  remove  the  effect  of  this  error  by  measuring  the 
inclination  of  the  horizontal  axis  to  the  horizon  and  applying  a  suitable  cor- 
rection to  the  circle  readings.  This  method  of  procedure  requires  a  knowl- 
edge of  the  theory  of  the  striding  level. 

The  striding  level  is  more  sensitive  than  the  plate  bubbles,  its  tube  is 
longer,  and  the  scale  includes  a  larger  number  of  divisions.  It  is  made  in  two 
forms,  one  with  the  zero  of  the  scale  at  the  middle  of  the  tube,  the  other  with 
the  zero  at  the  end.  Theoretically  the  two  forms  are  equivalent.  The  adjust- 
ment of  the  level  tube  within  its  mounting  should  be  such  that  the  bubble 
stands  at  the  middle  of  the  tube  when  the  base  line  is  horizontal.  The  scale 
reading  of  the  middle  of  the  bubble  for  this  position  is  called  the  horizontal 
reading.  Owing  to  residual  errors  of  adjustment,  the  horizontal  reading  will 


not  usually  be  zero,  even  for  the  form  in  which  the  zero  of  the  scale  is  at  the 
middle  of  the  tube.  Its  value  must  be  determined  and  applied  as  a  correc- 
tion to  the  scale  readings,  or  else  its  influence  must  be  eliminated.  The  latter 
is  easily  accomplished  by  combining  readings  made  in  the  direct  and  the 
reversed  position,  reversal  being  made  by  turning  the  level  end  for  end. 

Let  d=  the  angular  value  of  one  division  of  the  level  scale. 
Ji  =  the  horizontal  reading. 

Further,  for  any  inclination  of  the  base  line,  let  m'  and  m"  be  the  readings  of 
the  middle  of  the  bubble,  and  b'  and  b"  the  corresponding  observed  inclina- 
tions, for  the  level  direct  and  reversed,  respectively.  Finally,  assume  that  all 
readings  increasing  toward  the  right  are  positive,  and  all  toward  the  left,  neg- 
ative. We  then  find,  whatever  the  position  of  the  zero  of  the  scale, 

b'  =  (m'  —  Ji)d, 
b"  =  (m"  —  h}d. 

Since  h  has  opposite  signs  for  the  two  positions  of  the  level,  the  mean  of  (109) 
and  (no)  is 

b  =  Yi(m'  -f-  m"}d,  (ni) 

in  which  the  mean  of  the  observed  inclinations  has  been  written  equal  to  b. 
Denoting  by  r',  /',  and  r" ',  /",  the  readings  of  the  ends  of  the  bubble  for  two 
positions,  and  writing 

D  =  y±d,  (112) 

we  find  from  (HI) 

b  =  (r'  +  l'  +  r"  +  l"}D.  (113) 

This  result  depends  only  upon  the  readings  of  the  ends  of  the  bubble  and 
the  value  of  one  division  of  the  scale,  and  is  therefore  free  from  the  horizontal 
reading.  The  convention  regarding  the  algebraic  sign  is  such  that  when  b 
calculated  from  (113)  is  positive,  the  right  end  of  the  level  is  high. 

Since  b'  and  b"  are  two  observed  values  of  the  same  quantity,  we  find  from 
the  difference  of  (109)  and  (no) 

/i=X(r'  +  l'  —  r"  —  /"),  (114) 

which  may  be  used  for  the  calculation  of  h  when  a  complete  observation  has 
been  made. 

43.  Precepts  for  the  use  of  the  striding  level. — The  level  is  a  sen- 
sitive instrument,  and  great  care  must  be  exercised  in  its  manipulation  if  pre- 
cise results  are  to  be  obtained.  The  inclinations  to  be  measured  should  be 
small  and  the  horizontal  reading  should  correspond  as  closely  as  possible  with 
the  scale  reading  of  the  middle  of  the  tube.  The  points  of  contact  of  the  level 
with  the  pivots  upon  which  it  rests  must  be  carefully  freed  from  dust  particles. 


THE  LEVEL  73 

The  length  of  the  bubble,  which  is  adjustable  in  the  more  sensitive  forms, 
should  be  about  one-third  the  length  of  the  tube,  and  ample  time  should  be 
allowed  for  the  bubble  to  come  to  rest  before  reading.  The  instrument  should 
be  protected  from  changes  in  temperature,  and,  to  this  end,  it  should  be 
shielded  from  the  rays  of  the  sun,  and  from  the  heat  of  the  reading  lamp  and 
the  person  of  the  observer.  The  right  end  of  the  bubble  should  always  be 
read  first,  careful  attention  being  given  to  the  algebraic  sign,  and  the  time  of 
reversal  for  each  observation  should  be  noted.  Mistakes  in  reading  may  be 
avoided  by  noting  that  r'  —  /',  the  length  of  the  bubble,  must  equal  r"  —  /". 

The    following,    in    which  5  represents    the  sum  of  the  four  readings,  is  a 
convenient  form  for  the  record: 

Time 

r'  r' 

r"  r" 


r"  __    '          ,  —  Sd. 

S  is  most  easily  found  by  forming  first  the  diagonal  sums  of  the  four 
readings  written  as  above,  for  both  r'  and  /"  and  r"  and  /'  will  be  opposite  in 
sign  and  approximately  equal  in  absolute  magnitude. 

Example  28.  The  following  illustrates  the  record  and  reduction  of  level  observations. 
The  first  observation  was  made  with  a  level  whose  zero  point  is  in  the  middle  of  the  tube;  the 
second,  with  one  whose  zero  is  at  the  end.  The  values  of  D  are  S'.'i6  and  05032,  respectively. 

0  =  6*  15'"  r=9h  I2m 

+  14-1        -   9-7  +31-0      +  l6-4 

+  10.  r      —13.8  —20.3      —35-0 


44.  Determination  of  the  value  of  one  division  of  a  level.— The  ob- 
server should  be  familiar  with  the  sensitiveness  of  all  the  levels  of  his  instru- 
ment, even  though  he  depends  entirely  upon  a  simple  centering  of  the  bubble 
for  the  adjustment.  If  the  striding  level  is  to  be  used,  a  knowledge  of  the  an- 
gular value  of  one  division  of  its  scale  is  an  essential. 

The  investigation  of  levels  is  most  easily  carried  out  with  the  aid  of  a 
level  trier,  which  is  an  instrument  consisting  essentially  of  a  rigid  base  carry- 
ing a  movable  arm  whose  inclination  to  the  horizon  may  be  varied  by  a  known 
amount  by  means  of  a  graduated  micrometer  screw.  The  entire  transit  may 
be  mounted  on  the  arm,  or  the  various  levels  may  be  attached  separately  for 
the  investigation.  The  determination  of  the  change  in  the  inclination  of  the 
arm  of  the  level  trier  necessary  to  move  the  bubble  over  a  given  number  of 
divisions  gives  at  once  the  angular  value  of  one  division  of  the  scale. 


74 


PRACTICAL  ASTRONOMY 


Example  29.  The  following  shows  part  of  the  reduction  of  observations  made  with  a 
level  trier  for  the  determination  of  the  value  of  one  division  of  a  level.  The  bubble  was  run 
•from  the  left  to  the  right  end  of  the  tube  and  back  again,  for  both  level  direct  and  reversed, 
by  moving  the  micrometer  head  through  four  divisions  at  a  time.  The  ends  of  the  bubble 
were  read  for  each  setting  of  the  micrometer.  Column  two  of  the  table  gives  the  micrometer 
settings;  and  columns  three  and  four,  the  corresponding  means  of  the  end  readings  of  the 
bubble  for  level  direct.  The  fifth  column 'contains  the  means  of  the  quantities  in  the  two  pre- 
ceding columns;  and  column  six,  the  differences  between  the  wth  and  the  (6  +  w)th  readings 
in  column  five.  The  principle  used  in  combining  the  observations  is  the  same  as  that  em- 
ployed in  Examples  26  and  27.  The  length  of  the  arm  and  the  pitch  of  the  micrometer  screw 
are  such  that  a  rotation  of  the  micrometer  head  through  one  division  changes  the  inclination 
by  i".  Each  of  the  displacements  of  the  bubble  in  column  six  therefore  corresponds  to  a 
change  in  inclination  of  24".  The  quotients  formed  by  dividing  the  displacements  into  24" 
are  the  values  of  one  division  of  the  level  for  different  portions  of  the  tube.  A  similar  reduc- 
tion of  the  readings  taken  with  the  level  in  the  reversed  position  gave  for  d  the  values  in  column 
eight.  The  means  for  the  two  series  are  in  the  last  column.  A  glance  at  the  results  in  this 
column  is  sufficient  to  show  that  the  curvature  of  the  level  tube  is  variable. 

ONE  DIVISION  OF  A  LEVEL — LEVEL  TRIER 


No. 

Microm. 
Reading 

Reading  Middle  of  Bubble 

Dis- 
place- 
ment 

d 

L  to  R 

R  to  L 

Mean 

Direct 

Rev'sed 

Mean 

i 

164 

12.35 

12.95 

12.65 

H-45 

i  '.'66 

i  "73 

i  '.'70 

2 

160 

I5-.SO 

15-75 

15.62 

1333 

i.  So 

1.76 

i   78 

3 

156 

17.70 

18.00 

17-85 

I3-03 

1.84 

i  .84 

1.84 

4 

152 

19.80 

20.50 

20.15 

12.73 

1.89 

i  .90 

i  .90 

5 

148 

22.45 

23.00 

22.72 

12.30 

i-95 

i  .90 

1.92 

6 

144 

24-55 

25.20 

24.88 

12.22 

i  .96 

1.92 

1.94 

7 

140 

26.95 

27.25 

27.10 

8 

136 

28.90 

29.00 

28.95 

9 

132 

30.70 

31-05 

30.88 

10 

128 

32.60 

33-15 

32.88 

ii 

124 

34-95 

35-  10 

35-02 

12 

1  20 

37-05 

37-15 

37-io 

The  investigation  may  also  be  carried  out  by  a  method  first  proposed  by 
Comstock  in  which  the  circles  of  the  transit  are  used  to  change  the  inclination 
of  the  level  tube  by  a  known  amount.  If  the  instrument  is  levelled,  the  levels 
themselves  being  in  adjustment,  the  bubbles  will  remain  centered  when  the 
alidade  is  rotated  about  the  vertical  axis.  If  now  the  vertical  axis  be  deflected 
from  the  true  vertical  by  a  small  angle  ?',  the  bubbles  will  not  remain  centered 
as  the  instrument  is  rotated.  For  any  given  level,  however,  there  will  be  two 
readings  of  the  horizontal  circle,  differing  by  180°,  for  which  the  bubble  will 
stand  at  the  middle  of  the  tube;  and  by  rotating  slightly  about  the  vertical 
axis  it  can  be  brought  to  any  desired  position  in  the  tube.  The  change  in  the 
inclination  of  the  level  tube  corresponding  to  any  given  displacement  of  the 
bubble  can  be  expressed  in  terms  of  i  and  the  observed  change  in  the  reading 


ONE  DIVISION  OF  THE  LEVEL 


75 


of  the  horizontal  circle,  whence  the  angular  value  of  one  division  of  the  level 
may  be  determined  as  before. 

To  express  d  as  a  function  of  i  and  the  horizontal  circle  readings,  let  HC 
and  HC'  in  Fig.  9  represent  portions  of  the  horizontal  circles  for  the  normal 
and  the  deflected  positions  of  the  vertical  axis;  Z,  any  position  of  the  level, 
which  is  supposed  to  be  attached  with  its  axis  perpendicular  to  the  radius 
through  L  and  parallel  to  the  plane  of  the  circle;  and  b,  the  corresponding 
inclination.  In  the  spherical  right  triangle  HLC  the  angle  //is  equal  to  z,  the 


Fig.  9 

deflection  of  the  axis  from  the  vertical,  while  that  at  L  is  90°  —  b.  Now,  if  r0 
and  r  be  the  horizontal  circle  readings  corresponding  to  the  inclinations  zero 
and  b  respectively,  we  find 


Arc  HL  =  go°  —  (r  —  r0), 
whence  from  the  triangle  HLC, 

tan  #  =  tan  i  sin  (r  —  r0). 

The  angle  b  is  very  small  and,  for  i  equal  two  or  three  degrees,  r  —  ro  will 
never  exceed  one  degree.     We  may  therefore  use  the  approximate  relation 

d  =  (r  —  r0)  tan  2,  (n6) 

with  an  error  not  exceeding  o"oi. 

For  any  other  inclination,  b^  we  have  the  analogous  equation 


=(r\  —  r0)  tan  z, 


which,  combined  with  (112)  gives 


The  angle  r^  —  r  is  the  change  in  the  horizontal  circle  reading  correspon- 
ding to  the  change  in  inclination  b±  —  b.  The  latter,  however,  may  be  written 
equal  to  sdt  where  s  is  the  displacement  of  the  bubble  in  scale  divisions,  and  d 
the  angular  value  of  one  division. 

We  thus  have  finally  as  the  expression  for  d 


d  = 


tan  ^. 


(118) 


76 


PRACTICAL  ASTRONOMY 


The  angle  i  should  be  two  or  three  degrees  for  the  investigation  of  the 
ordinary  transit  levels.  For  very  sensitive  levels  it  should  be  less.  If  the 
instrument  be  provided  with  a  telescope  level,  the  deflection  of  the  vertical 
axis  may  be  accomplished  as  follows:  Level  the  instrument  and  center  the 
telescope  bubble.  Then  change  the.vertical  circle  reading  by  the  angle  i  and, 
by  means  of  the  levelling  screws,  bring  the  telescope  bubble  back  to  the 
middle  of  its  tube,  taking  care  at  the  same  time  that  the  transverse  plate 
bubble  is  also  centered  after  the  deflection.  This  precaution  is  necessary  in 
order  that  the  deflection  may  have  no  component  perpendicular  to  the  plane 
of  the  vertical  circle.  In  the  absence  of  a  telescope  level,  level  the  instrument, 
sight  on  a  distant  object,  change  the  vertical  circle  reading  by  i,  and  bring  the 
object  back  to  the  intersection  of  the  threads  by  means  of  the  levelling  screws. 

The  observations  may  be  made  either  by  displacing  the  bubble  through  a 
definite  number  of  divisions  and  noting  the  corresponding  change  in  the  hori- 
zontal circle  readings,  or  by  changing  the  circle  readings  by  a  definite  amount, 
say  10',  and  observing  the  variations  in  the  position  of  the  bubble.  For  short 
tubes  with  only  a  few  graduations  the  former  method  is  more  convenient,  while 
the  latter  is  to  be  preferred  for  the  long  finely  graduated  tubes  of  sensitive 
levels. 

The  bubble  should  be  run  from  one  end  of  the  tube  to  the  other  and  then 
back  again,  in  both  positions  of  the  instrument.  Such  a  series  of  readings 
constitutes  a  set. 

The  instrument  must  be  as  rigidly  mounted  as  possible,  preferably  on  a 
masonry  pier.  It  is  desirable  to  check  the  constancy  of  i  by  deflecting  through 
this  angle  toward  the  vertical  at  the  end  of  a  set  and  noting  whether  the 
instrument  is  then  levelled. 

Example  30.  Observations  were  made  by  the  deflected  axis  method  for  the  determina- 
tion of  the  value  of  one  division  of  the  striding  level  of  a  Berger  transit.  The  deflection  was 
3°.  The  graduations  of  the  tube  are  in  two  groups  of  three  each,  the  groups  being  separated 

ONE  DIVISION  OF  A  LEVEL — DEFLECTED  Axis 


Level 
Divisions 

Hor.  Circle  Reading 

Mean 

r^  —  r 

Position 

L  to  R 

R  to  L 

i  and  4 
3  and  6 

34i°      55  .'o 
342         8.0 

34i°     56.'o 
342         8.0 

5  5  -'5 
8.0 

1  2'.  5 

Direct 
Direct 

3  and  6 
i  and  4 

162         6.5 
161       53.0 

162         6.3 
161        53.0 

6.4 

53-o 
Mean 

13-4 

Reversed 
Reversed 

12-95 

»=    3U 

—  r=  12/95 

S=      2 


tan  i  =  8.719 

i  —  f")=  1.  112 

colog  5  ;=  9.699 

log^=  9.530 


A  us. 


MEASUREMENT  OF  ALTITUDE  77 

by  a  space  approximately  equal  to  the  length  of  the  bubble.  The  horizontal  circle  was  read 
when  the  bubble  was  symmetrically  placed  with  respect  to  the  pairs  of  graduations  indicated 
in  column  one  of  the  table.  The  circle  readings  themselves  are  in  columns  two  and  three; 
and  the  minutes  of  the  means  of  corresponding  settings,  in  column  four.  The  differences  of 
the  readings  for  a  displacement  of  the  bubble  through  two  divisions  are  in  the  fifth  column. 
The  calculation  for  the  determination  of  d  is  in  accordance  with  equation  (118). 

45.  The  measurement  of  vertical  angles. — The  observer  will  have 
occasion  to  measure  the  altitude  not  only  of  rapidly  moving  equatorial  stars 
but  also  of  circumpolar  objects  like  Polaris  whose  positions  with  respect  to  the 
horizon  change  but  slowly.  The  difference  in  motion  in  the  two  cases  neces- 
sitates a  difference  of  method  in  making  the  settings.  For  Polaris  or  any  other 
close  circumpolar  object,  the  star  should  be  brought  to  the  intersection  of  the 
threads  by  the  slow  motions,  the  time  of  coincidence  and  the  vertical  circle 
readings  being  carefully  noted.  For  stars  whose  altitude  varies  rapidly,  this 
cannot  be  done  with  precision.  The  object  is  therefore  brought  into  coinci- 
dence with  the  vertical  thread  near  the  point  of  intersection,  and  kept  on  the 
thread  by  slowly  turning  the  horizontal  slow  motion  until  the  instant  of 
transit  across  the  horizontal  thread,  the  time  and  the  vertical  circle  readings 
being  noted  as  before. 

Observations  on  the  sun  are  most  readily  made  with  the  aid  of  a  shade  of 
colored  glass,  but  if  this  is  not  available,  the  image  may  be  projected  on  a  card 
held  a  few  inches  back  of  the  eyepiece,  by  a  proper  focusing  of  the  objective. 
In  order  that  the  threads  may  be  seen  sharply  defined  on  the  card,  it  is  neces- 
sary that  the  eyepiece  be  drawn  out  a  small  fraction  of  an  inch  from  its 
normal  position  before  the  solar  image  is  focused.  There  are  several  methods 
by  which  the  pointings  may  be  made.  For  example,  the  instrument  may  be 
adjusted  so  that  the  preceding  limb  is  near  the  horizontal  thread  and  ap- 
proaching the  intersection.  The  instrument  is  clamped  and  the  instant  of 
tangency  carefully  noted.  Then,  without  changing  the  vertical  circle  reading, 
the  image  is  allowed  to  trail  through  the  field  until  the  transit  of  the  following- 
limb  occurs,  when  the  time  is  again  noted,  -the  instrument  in  the  meantime 
being  rotated  by  means  of  the  horizontal  slow  motion  so  that  both  transits  are 
observed  at  the  intersection  of  the  threads.  While  waiting  for  the  second 
transit,  the  vertical  circle  is  read.  This  method  is  open  to  the  objection  that 
an  interval  of  three  or  four  minutes  separates  the  transits  of  the  two  limbs, 
which  entails  a  considerable  loss  of  time.  The  interval  may  be  shortened  by 
shifting  the  position  of  the  telescope  between  the  observations,  but  this  of 
course  requires  a  reading  of  the  vertical  circle  for  each  transit.  If  there  be 
more  than  one  horizontal  thread,  the  difficulty  can  be  avoided  by  observing 
the  transits  over  the  extreme  threads — the  preceding  limb  over  the  first 
thread  and  the  following  limb  over  the  last  thread.  The  same  number  of 
settings  should  be  made  for  both  limbs.  The  mean  of  the  readings  will  then 
correspond  to  the  altitude  of  the  sun's  center,  the  influence  of  semidiameter 
being  eliminated.  If  for  any  reason  the  program  cannot  be  made  complete 
in  this  particular,  the  altitude  of  the  sun's  center  may  still  be  found  with  the 


78  PRACTICAL  ASTRONOMY 

aid  of  the  value  of  the  semidiameter  interpolated  from  page  I  of  the  Epliemeris 
for  the  instant  of  observation. 

The  arrangement  of  the  observing  program  is  determined  by  the- results 
derived  in  Section  40  and  summarized  in  Section  41.  The  number  of  settings 
to  be  made  for  the  determination  of  the  altitude  depends  upon  the  precision 
desired,  the  rapidity  with  which  the  observer  can  make  the  pointings  and  read 
the  circle,  and  the  position  of  the  object.  It  is  desirable,  however,  that  the 
number  should  not  be  less  than  two  for  each  position  of  the  instrument.  The 
maximum  number  to  be  included  in  a  single  set  is  limited  by  the  fact  that  it  is 
convenient  to  use  for  the  reduction  the  means  of  the  circle  readings  and  the 
times.  Since  the  change  in  the  altitude  of  the  star  is  not  proportional  to  the 
change  in  the  time,  the  two  means,  rigorously  speaking,  will  not  correspond  to 
each  other;  but  if  the  observing  interval  does  not  exceed  a  certain  limit,  say  a 
quarter  of  an  hour,  no  appreciable  error  will  be  introduced  into  results  secured 
with  the  engineer's  transit  by  treating  the  means  as  a  single  observation.  The 
observing  program  will  also  depend  on  the  method  employed  for  the  elimina- 
tion of  the  instrumental  errors  i,j,  c  and  /.  Bearing  in  mind  the  various  fac- 
tors involved,  we  adopt  the  following  as  convenient  arrangements  for  a  set  of 
observations  on  a  star.  The  necessary  modifications  for  measures  on  the  sun 
will  at  once  be  suggested  by  the  methods  for  making  the  settings  described  in 
the  preceding  paragraph. 

OBSERVATIONS  DIRECT  OBSERVATIONS  DIRECT  AXD  REFLECTED 

Level.  Level. 

2  readings  on  star,  C.  R.  i   reading  on  star,  direct.        ^ 

Reverse  2  readings  on  star,  reflected,  I  C.R. 

Level.  i   reading  on  star,  direct.        J 

4  readings  on  star,  C.  L.  Reverse. 

Reverse.  i  reading  on  star,  direct.        % 

Level.  2  readings  on  star,  reflected.  j-C.L. 

2  Readings  on  star,  C.  R.  i  reading  on  star,  direct.        J 

With  the  first  arrangement,  which  is  to  be  used  when  all  of  the  pointings 
are  made  directly  on  the  star,  the  elimination  of  the  errors  depends  upon  the 
bubbles  occupying  the  same  positions  in  their  tubes  for  both  C.R.  and  C.L. 
The  instrument  must  therefore  be  relevelled  carefully  after  each  reversal. 

With  the  second,  which  will  find  application  when  the  artificial  horizon  is 
employed,  the  elimination  will  be  complete  if  the  adjustments  remain  un- 
changed during  the  intervals  separating  the  various  direct  observations  and 
the  corresponding  reflected  observations  immediately  preceding  or  following. 
After  the  instrument  has  once  been  levelled,  therefore,  the  screws  need  not  be 
touched  until  the  set  has  been  completed  unless  the  bubbles  should  become 
displaced  by  a  considerable  amount. 

Both  verniers  should  be  read  for  each  setting  of  the  telescope. 

If  only  an  approximate  result  is  required,  the  observations  may  be  dis- 
continued at  the  middle  of  the  set.  On  the  other  hand,  if  more  precision  is 


MEASUREMENT  OF  ALTITUDE  79 

desired,  additional  sets  may  be  observed,  each  of  which,  however,  should  be 
reduced  separately. 

The  fact  that  for  a  short  interval  the  change  in  the  altitude  is  sensibly 
proportional  to  the  change  in  the  time  makes  it  possible  to  test  the  consist- 
ency of  the  measures.  For  direct  observations  the  quotients  of  the  differ- 
ences between  the  successive  circle  readings  by  the  differences  between  the 
corresponding  times  must  be  sensibly  equal.  If  this  condition  is  not  satis- 
fied, an  error  has  been  committed.  The  errors  most  likely  to  occur  are 
those  involving  mistakes  of  10'  or  20',  or  perhaps  a  whole  degree,  in  the 
circle  readings,  and  an  exact  number  of  minutes  in  the  times.  It  is  convenient 
to  express  differences  of  the  circle  readings  in  minutes  of  arc,  and  the  time 
intervals  in  minutes  and  tenths.  The  quotients  will  thus  express  the  change 
in  the  altitude  in  minutes  of  arc  for  one  minute  of  time.  If  the  artificial 
horizon  has  been  used  the  quotients  must  be  calculated  for  the  direct  and 
reflected  observations  separately.  For  observations  on  the  sun,  the  combina- 
tion of  the  data  for  the  calculation  of  the  quotients  will  depend  upon  the 
method  followed  in  making  the  settings,  and  is  easily  derived  in  any  special 
case.  The  test  is  usually  sufficient  to  locate  errors  of  the  class  mentioned 
with  such  certaint)'  as  to  justify  a  correction  of  the  original  record,  and  should 
always  be  applied  immediately  after  the  completion  of  the  set  in  order  that 
the  measures  may  be  repeated  if  necessary.  For  circumpolar  objects,  a  simple 
inspection  will  usually  be  sufficient  to  indicate  the  consistency  of  the  observa- 
tions. 

Equations  (103)  and  (104),  and  (107)  show  that  for  an  instrument  graduated 
to  read  altitudes,  the  apparent  altitude,  free  from  the  instrumental  errors,  i,j\ 
c,  and  /,  will  be  given  by  forming  the  mean  of  the  circle  readings  obtained  in 
accordance  with  the  above  programs. 

For  an  instrument  with  its  vertical  circle  graduated  continuously  o°  to 
360°  the  zenith  distance  will  be  given  by 

z0=%(vl  —  v^)  (119) 

where  the  subscripts  are  assigned  in  such  a  manner  that  vt  —  v2  <  180°.  If  the 
observations  are  direct,  one  v  will  represent  the  mean  of  all  the  circle  readings 
C.R.;  the  other,  the  mean  of  all  C.L.  If  the  artificial  horizon  has  been  used, 
one  v  will  represent  the  mean  of  all  the  direct  readings;  and  the  other,  the 
mean  of  all  the  reflected  readings. 

The  observed  altitude,  or  zenith  distance,  thus  derived  must  be  corrected 
for  refraction  and  parallax  in  accordance  with  Sections  8  and  9. 

Example  31.  The  following  is  the  record  of  partial  sets  of  observations  made  with  a 
Buff  &  Buff  engineer's  transit  at  the  Laws  Observatory,  on  1908,  Oct.  2,  Friday  P.  M.,  for  the 
determination  of  the  altitudes  of  Polaris  and  Alcyone.  The  measures  were  all  direct.  The 
timepiece  used  was  an  Elgin  watch. 

An  inspection  of  the  readings  for  Polaris  shows  that  the  measures  are  consistent.  The 
relatively  large  difference  in  the  readings  C.  R.  and  C.  L.  reveals  the  existence  of  an  index 
error  of  2'  or  3'. 


80 


PR  A  C  TIC  A  L  ASTR  ONOM1 ' 


POLARIS 

Watch 

Vertical 
Ver.   A 

Circle 
Ver.    B 

Circle 

Sh3i* 

"19" 

39° 

26' 

39° 

26' 

R 

33 

25 

26 

26 

R 

36 

50 

33 

33 

L 

36 

ii 

34 

34 

L 

Sh  35" 

,„• 

39° 

29'. 

8 

Watch 

ALCYONE 

Vertical  Circle 
Ver.  A      Ver.  B 

Circle    Rate 

h35m2i?4 

20° 

40' 

20° 

40' 

L 

37      1.2 

2O 

59 

20 

59 

L 

I  1/2 

41      1.4 

21 

38 

21 

3« 

R 

43      1-4 

22 

I 

22 

i 

R 

"•5 

21°   19:5 

For  Alcyone  the  close  agreement  of  the  values  for  the  rate  of  change  in  altitude  per 
minute  of  time  given  in  the  last  column  is  evidence  of  the  consistency  of  the  measures. 

The  quantities  in  the  fifth  line  are  the  means.  The  angles  are  the  apparent  altitudes 
corresponding  to  the  watch  times  immediately  preceding.  To  obtain  the  true  altitudes  a 
correction  for  refraction,  which  may  be  obtained  from  Table  I,  page  20,  must  be  applied. 

Example  32.  The  following  observations  were  made  with  a  Berger  engineer's  transit 
on  1908,  October  J5,  Thursday  P.  M.,  for  the  determination  of  the  altitude  of  the  sun.  The 
measures  were  all  direct  and  were  made  by  projecting  the  image  of  the  sun  on  a  card.  The 
transits  were  observed  over  the  middle  horizontal  thread,  the  telescope  being  shifted  after 
each  transit.  The  timepiece  was  the  Fauth  sidereal  clock  of  the  Laws  Observatory. 

Vertical    Circle 


Fauth 

Clk. 

Ver.  A 

Ver.  B 

Limb 

Circle 

Rate 

[6h37iri 

'5' 

!2 

24° 

l' 

24° 

r 

F 

R 

40 

46 

.8 

23 

33 

23 

33 

F 

R 

9/6 

43 

34 

•3 

22 

35 

22 

35 

P 

R 

45 

19 

.1 

22 

19 

22 

19 

P 

R 

9-2 

47 

7 

•5 

22 

26 

22 

27 

P 

L 

48 

47 

•3 

22 

ii 

22 

1  1 

F 

L 

9.0 

50 

1  1 

•7 

21 

25 

21 

25 

P 

L 

IO.O 

5i 

29 

•9 

21 

12 

21 

12 

P 

L 

Means      i6h  45'"  3855 


27'.8 


46.  The  measurement  of  horizontal  angles. — It  is  assumed  that 
the  two  objects  whose  difference  of  azimuth  is  to  be  determined  are  a  terres- 
trial mark  and  a  celestial  body,  either  the  sun  or  a  star.  The  directions  given 
in  Section  45  for  making  settings  in  the  measurement  of  vertical  angles  apply 
here  with  only  slight  and  obvious  modifications.  The  conditions  determining 
the  arrangement  of  the  observing  program  are  similar  to  those  enumerated  in 
the  present  section.  Although  the  details  may  vary  with  circumstances,  the 
following  will  serve  to  indicate  the  essentials.  The  first  arrangement  is  in- 
tended for  use  when  only  an  approximate  result  is  required,  while  the  second 
and  third  are  designed  for  more  precise  determinations.  The  first  two  include 
only  direct  observations,  while  the  last  is  arranged  for  measures  in  which  the 
artificial  horizon  is  employed.  In  direct  observations  care  should  be  taken  to 
keep  the  bubbles  centered  throughout,  but  when  the  artificial  horizon  is  used, 
the  levelling  screws  must  not  be  touched  between  any  direct  observation  and 
its  corresponding  reflected  setting.  For  settings  on  the  mark  the  zenith  dis- 
tance will  usually  be  so  nearly  equal  to  90°  that  the  error  due  to  the  deviation 
of  the  vertical  axis  from  the  true  vertical  will  be  quite  insensible,  even 
though  no  special  effort  be  made  to  eliminate  its  influence. 


HORIZONTAL  ANGLES 


81 


DIRECT  OBSERVATIONS 

1  setting  on  mark  \ 

2  settings  on  star  / 
2  settings  on  star  ~\ 
I  setting  on  mark  / 


DIRECT  OBSERVATIONS 
i  setting  on  mark  C.R. 
i  setting  on  mark  C.L. 
3  settings  on  star  C.L. 
3  settings  on  star  C.R. 
i  setting  on  mark  C.R. 
i  setting  on  mark  C.L. 


DIRECT  AND  REFLECTED  OBSERVATIONS 
i  setting  on  mark  C.R. 


i  setting  on  mark  C.L. 

i  setting  on  star,    direct       \ 

i  setting  on  star,    reflected  / 

i  setting  on  star,    reflected  \  ,„ 

i  setting  on  star,    direct       / 

i  setting  on  mark  C.R. 

i  setting  on  mark  C.L. 

Both  verniers  of  the  horizontal  circle  should  be  read  for  each  setting,  and 
for  those  made  on  the  star,  the  time  should  be  noted  in  addition. 

The  required  difference  of  azimuth  will  be  the  difference  between  the 
means  of  the  readings  on  the  mark  and  on  the  star.  Its  value  will  correspond 
to  the  mean  of  the  times.  If  more  precision  is  desired  than  can  be  obtained 
from  a  single  set,  several  sets  may  be  observed,  each  of  which  should  be  re- 
duced separately.  To  reduce  the  influence  of  graduation  error,  the  horizontal 
circle  should  be  shifted  between  the  sets.  If  the  number  of  sets  is  n,  the 
amount  of  the  shift  between  the  successive  sets  should  be 


Example  33.   The  following  is  the  record  of  a  simultaneous  determination  of  the  altitude 
of  Polaris  and  the  difference  in  azimuth  of  Polaris  and  a  mark. 


ALTITUDE  OF  POLARIS  AND  AZIMUTH  OF  MARK  No.  2 

1908,  Oct.  13,  Thursday  P.  M. 
Station  No.  2 

Buff  &  Buff  Engineer's  Transit  No.  5606 


Observer  Sh. 
Recorder  W. 


Object         Watch 

•57  at  7h  59m  P.M.,  and  — 
Hor.  Circle 

Ver.  A              Ver.  B 
147°   23/5         327°   23/0 

3154  at  9h  54m  P.M. 
Vertical  Circle 
Ver.  A       Ver.  B 

Circle 
R 

iviarK 

Polaris      9h 

35m 

23" 

322 

10 

.0 

142 

10 

.0 

39°  5i'      39°  5i' 

R 

Polaris 

40 

35 

8. 

o 

8. 

o 

51              51 

R 

Polaris 

44 

33 

142 

6 

•5 

322 

6 

•5 

59              59 

L 

Polaris 

48 

S 

4 

5 

5 

0 

59              58 

L 

TVT  n  r  lr 

327 

23 

•5 

147 

23 

.0 

L 

iTA  til  IV 

Means       gh 

42- 

10s 

/Star 
\  Mark 

322° 

7-' 

23' 

31  1 
25  / 

39°  54-'9  =  APPt.  Alt. 

Difference  of  Azimuth 

S-; 

]/= 

=  174 

44- 

06 

47.  The  method  of  repetitions. — The  precision  of  the  measurement  of 
the  azimuth  difference,  D,  of  two  objects,  A  and  B,  may  be  increased  materi- 
ally by  making  a  series  of  alternate  settings  on  A  and  B  such  that  the  rotation 
from  A  to  B  is  always  made  with  the  upper  motion  of  the  instrument,  and  that 
from  B  to  A  with  the  lower  motion.  Assuming  that  the  graduations  of  the 
horizontal  circle  increase  in  the  direction  AB,  each  turning  from  A  to  B  will 
6 


82  PRACTICAL  ASTRONOMY 

increase  the  reading  by  the  angle  D,  while  that  from  B  back  to  A  will  produce 
no  change  since  during  this  rotation  the  vernier  remains  clamped  to  the  circle. 
If  the  turning  from  A  to  B  is  repeated  n  times,  the  difference  between  the 
circle  readings  for  the  final  setting  on  B  and  the  initial  setting  on  A  will  be 
nD ;  and  if  the  initial  and  final  readings  be  R,  and  R2,  repectively,  we  shall 
have 

7->  R*  -^I 

D=-         — .  (120) 

n 

The  method  of  repetitions  derives  its  advantage  from  the  fact  that  the 
circle  is  not  read  for  the  intermediate  settings  on  A  and  B.  Not  only  is  the 
observer  thus  spared  considerable  labor,  but,  what  is  of  more  importance,  the 
errors  which  necessarily  would  affect  the  readings  do  not  enter  into  the  result. 
Consequently,  that  part  of  the  resultant  error  of  observation  arising  from  the 
intermediate  settings  is  due  solely  to  the  imperfect  setting  of  the  cross  threads 
on  the  object.  For  instruments  such  as  the  engineer's  transit,  in  which  the 
uncertainty  accompanying  the  reading  of  the  angle  is  large  as  compared  with 
that  of  the  pointing  on  the  object,  the  precision  of  the  result  given  by  (120) 
will  be  considerably  greater  than  that  of  the  mean  of  n  separate  measurements 
of  the  angle  D,  each  of  which  requires  two  readings  of  the  circle.  But  for 
instruments  in  which  the  accuracy  of  the  readings  is  comparable  with  that  of 
the  pointings,  as  is  the  case  with  the  modern  theodolite  provided  with  read- 
ing microscopes,  the  method  of  repetitions  is  not  to  be  recommended. 
Although  there  is  even  here  a  theoretical  advantage,  it  is  offset  by  the  fact 
that  the  peculiar  observing  program  required  for  the  application  of  the  method 
presupposes  the  stability  of  the  instrument  for  a  relatively  long  interval,  and 
hence  affords  an  unusual  opportunity  for  small  variations  in  position  to  affect 
the  precision  of  the  measures.  Moreover,  experience  has  shown  that  there 
are  small  systematic  errors  dependent  upon  the  direction  of  measurement,  i.e. 
upon  whether  the  initial  setting  is  made  on  A  or  on  B;  and,  although  these 
may  be  eliminated  in  part  by  combining  series  measured  in  opposite  direc- 
tions, it  is  not  certain  that  the  compensation  is  of  the  completeness  requisite 
for  observations  of  the  highest  precision.  With  the  engineer's  transit,  how- 
ever, the  method  of  repetitions  may  be  used  with  advantage. 

Since  rotation  takes  place  on  both  the  upper  and  the  lower  motions,  any 
non-parallelism  of  the  vertical  axes  will  affect  the  readings;  and  the  observing 
program  must  be  arranged  to  eliminate  this  along  with  the  other  instru- 
mental errors.  For  any  given  setting  the  deviation  of  the  axis  from  parallel- 
ism,/, unites  with  the  inclination  of  the  lower  axes  to  the  true  vertical,  i' ,  and 
determines  the  value  of  z,  the  inclination  of  the  upper  axis  to  the  vertical,  for 
the  setting  in  question.  For  different  settings  i  will  be  different,  for  a 
rotation  of  the  instrument  on  the  lower  motion  causes  the  upper  axis  to 
describe  a  cone  whose  apex  angle  is  2p  and  whose  axis  is  inclined  to  the  true 
vertical  by  i'.  But  no  matter  what  the  magnitude  of  i  may  be,  within  certain 
limits  easily  including  allvalues  arising  in  practice,  it  may  be  eliminated  by 
forming  the  mean  of  direct  and  reflected  readings  made  in  the  same  position 


METHOD  OF  REPETITIONS  83 

of  the  instrument,  provided  that  i  is  the  same  in  direction  and  magnitude  for 
both  settings.  This  follows  from  the  discussion  on  pages  66  and  67  whose 
result  is  expressed  by  equation  (88).  Hence,  if  after  a  series  of  n  repetitions 
observed  C.R.  direct,  n  further  repetitions  be  made  C.R.  reflected,  such  that 
the  vernier  readings  for  the  corresponding  settings  in  the  two  series  are 
approximately  the  same,  the  instrumental  errors  i'  and  /will  be  eliminated. 
Equation  (88)  shows  thaty,  the  deviation  of  the  upper  vertical  axis  from  per- 
pendicularity with  the  horizontal  axis,  will  also  be  eliminated.  To  remove 
the  influence  of  the  collimation,  c,  the  entire  process  must  be  repeated  C.L.; 
and  to  neutralize  the  systematic  error  dependent  upon  the  direction  of  meas- 
urement, the  direct  and  reflected  series  should  be  measured  in  opposite  direc- 
tions. We  thus  have  the  following  observing  program,  in  which  A'  and  B' 
denote  the  reflected  images  of  A  and  B,  respectively: 

Level  on  the  lower  motion. 
rSet  on  A  and  read  the  horizontal  circle. 
Direct         -j  Turn  from  A  to  B  on  the  upper  motion  n  times. 

I  Read  the  horizontal  circle  for  last  setting  on  B. 

j-C.R. 
i  Set  on  B'  and  read  the  horizontal  circle. 

Reflected    <  Turn  from  B'  to  A'  on  the  upper  motion  n  times. 
'  Read  the  horizontal  circle  for  last  setting  on  A'. 
Repeat  for  C.L. 

The  circle  reading  for  the  first  setting  on  B'  must  be  the  same,  approximately  at  least, 
as  that  for  the  last  setting  on  B, 

The  mean  of  the  values  of  D  calculated  from  the  four  series  is  the  required  azimuth 
difference  of  A  and  B. 

Uusually  one  of  the  objects,  say  A,  will  be  near  the  horizon,  in  which  case 
reflected  settings  on  A'  will  be  impossible.  A  must  then  be  substituted  for 
A'  in  the  above  program.  The  error  due  to  i  will  not  be  eliminated  from  these 
settings;  but,  owing  to  the  presence  of  the  factor  cot£0,  it  may  be  disregarded. 

When  the  artificial  horizon  is  not  used    the    program    must   be   modified. 
Were  i'  zero,  /would  constantly  be  equal  to/,  although  the    direction    of   the 
deflection  would  change   with    a   rotation    of   the    instrument   on    the    lower 
motion.     If  a  series  of  n  repetitions  C.R.  be  made  under  these   circumstances,, 
equation  (82)  shows  that  each  setting  will  be  affected  by  an  error  of  the  form 

/cot£0  +  />  cos /cot  z0  -f-  c  cosec,sr0. 

The  first  and  last  terms  of  this  expression  will  have  the  same  values  for  all' 
pointings  on  the  same  object.  Equations  (82),  (84),  and  (86)  show  that  they  may 
be  eliminated  by  combining  with  a  similar  series  made  C.L.  The  values  of 
the  second  term  will  be  different  for  each  setting  owing  to  the  change  in  /,  but 
their  sum  will  be  zero  if  the  values  of  /  are  uniformly  distributed  throughout 
360°,  or  any  multiple  of  360°.  In  order  that  this  may  be  the  case,  approxi- 
mately at  least,  it  is  only  necessary  that  n  be  the  integer  most  nearly  equaling, 
kT)6o0/D,  where  the  k  is  any  integer,  in  practice  usually  i  or  2. 

It  is  also  easily  seen  that,  if  after  any  arbitrary  number  of  settings  the 
instrument  be  reversed  about  the  lozver  motion  and  the  series  repeated  in.  the 


84  PRACTICAL  ASTRONOMY 

reverse  order,  the  sum  of  the  errors  involving/  will  be  zero,  provided  that  the 
circle  readings  for  corresponding  settings  C.R.  and  C.L.  are  the  same,  or  ap- 
proximately so.  The  reversal  of  the  instrument  on  the  lower  motion  changes 
the  direction  of  the  deflection/  by  180°.  The  values  of  /  for  corresponding 
settings  C.R.  and  C.L.  will  therefore  differ  by  180°,  and  the  errors  will  be  oppo- 
site in  sign  and  will  cancel  when  the  mean  of  the  two  series  is  formed.  The 
reversal  also  eliminates  the  influence  of  j  and  c  as  indicated  in  the  preceding 
paragraph. 

The  above  assumes  that  the  deflection  of  the  lower  axis,  /',  is  zero.  If 
this  is  not  the  case,  each  setting  will  be  affected  by  an  additional  error  of  the 
form  i'  cos  /'  cot  z0,  in  which  /'  is  constant  so  long  as  i'  remains  unchanged 
in  direction.  If  i'  be  the  result  of  a  non-adjustment  of  the  plate  bubbles,  the 
error  which  it  produces  may  be  eliminated  from  the  mean  of  two  series,  one 
C.R.  and  one  C.L.,  by  relevelling  after  reversal.  (See  page  70.)  This  will 
change  the  direction  of  i'  by  180°.  Consequently,  the  values  of  /'  for  C.R. 
and  C.L.  will  differ  by  180°,  and  the  errors  for  the  two  positions  will  neutralize 
each  other  when  the  mean  is  formed. 

The  consideration  of  these  results  leads  to  the  following  arrangement  of 
the  observing  program. 

Level  on  the  lower  motion. 

Set  on  A  and  read  horizontal  circle.  ^ 

Turn  from  A  to  B  on  upper  motion  «  times.     \  C.R. 

Read  horizontal  circle  for  last  setting  on  B. 

Reverse  on  lower  motion  and  relevel. 

Set  on  B  and  read  horizontal  circle.  \ 

Turn  from  B  to  A  on  upper  motion  n  times.     I  C.L. 

Read  horizontal  circle  for  last  setting  on  A.     J 

The  circle  reading  for  the  first  setting  on  B,  C.L.  should  be  the  same,  approximately  at 
least,  as  that  for  the  last  setting  on  B,  C.R. 

The  mean   of  the  values  of  D  calculated  from   the  two   series  is   the  required  azimuth 
difference  of  A  and  B. 

With  this  arrangement  the  instrumental  errors  i',p,j,  and  c  will  be  com- 
pletely eliminated,  whether  the  settings  are  distributed  through  360°  or  not, 
provided  only  that  the  instrumental  errors  remain  constant  during  the  obser- 
vations. Practically,  it  is  desirable  that  the  value  of  n  should  be  such  that  nD 
equals  360°,  or  a  multiple  of  360°,  at  least  approximately;  but  when  D  is  small 
this  may  unduly  prolong  the  observations.  The  maximum  number  of  repeti- 
tions which  can  be  made  advantageously  depends  upon  the  stability  of  the 
instrument  and  must  be  determined  by  experience. 

If  the  instrument  is  provided  with  a  striding  level,  the  influence  of  z',  /, 
and  /  may  be  taken  into  account  by  measuring  the  inclination  of  the  horizontal 
axis  for  each  setting  and  applying  a  correction  to  Rl  and  Ra  of  the  form  b  cot  z0, 
in  which  b  denotes  the  sum  of  all  the  observed  inclinations  for  settings  on 
A  and  B  respectively. 

When  one  of  the  objects,  say  B,  is  a  star,  the  time  of  each  setting  on  B 
must  be  noted.  The  calculated  value  of  D  will  then  correspond  sensibly  to 
the  mean  of  the  times,  provided  the  observing  program  be  not  too  long. 


THE  SEXTANT 


85 


Example  34.  On  1909,  April  9,  the  following  observations  of  the  difference  in  azimuth 
of  Polaris  and  a  mark  were  made  by  the  method  of  repetitions  with  a  Buff  &  Buff  engineer's 
transit.  The  recorded  times  are  those  of  a  Fauth  sidereal  clock  whose  error  was  -}-6m36B. 
After  four  repetitions  C.R.,  the  instrument  was  reversed  on  the  lower  motion,  relevelled,  and 
the  series  repeated  in  the  reverse  order.  Since  the  azimuth  difference  is  approximately  174°, 
720°  must  be  added  to  the  readings  on  the  star  before  combining  them  with  those  on  the 
mark.  The  results  for  the  two  halves  are  derived  separately,  although  the  means  for  the  set 
are  also  given. 


Hor. 

Circle 

Object 

0F 

Ver.  A 

Ver.  B 

Circle 

Means 

Mark         





179°  59'-  5 

359°   59'-  5 

R 

179° 

59' 

30" 

Polaris     9h 

27™ 

3°" 

353     32 

R 

Polaris 

3° 

48 

R 

Polaris 

33 

36 

R 

Polaris 

35 

49 

i  54     13  •  5 

334     13-5 

R 

874 

13 

3° 

4) 

7 

43 

4)694 

i4 

o 

9 

3i 

56 

0  =  9h 

3Sm  338             .. 

?-M 

=  173 

33 

30 

Polaris     9 

39 

12 

154     i3-o 

334     13-5 

L 

874 

13 

15 

Polaris 

42 

28 

L 

Polaris 

44 

9 

L 

Polaris 

46 

20 

L 

Mark        





179     51.0 

359    5i-o 

L 

179 

5i 

o 

4) 

12 

9 

4)694 

22 

15 

9 

43 

2 

0  =  9* 

49m  3S8 

S"  —  M 

=  173 

35 

34 

Final  Means 

0  =  9 

44       5 

S  —  M 

=  173 

34 

32 

THE    SEXTANT 

48.  Historical  and  descriptive. — The  instruments  typified  by  the  engi- 
neer's transit  may  be  used  for  the  measurement  of  horizontal  or  vertical  angles 
only.  Simultaneously  with  the  development  of  the  altazimuth  principle  there 
was  gradually  evolved  a  contrivance  adapted  for  the  measurement  of  angles 
lying  in  any  plane.  Beginning  with  the  astrolabe  of  the  ancients,  the  applica- 
tion of  various  ideas  gave  in  succession  the  Jacob's  staff,  or  cross-staff,  which 
dates  apparently  from  the  middle  of  the  fourteenth  century,  the  back-staff,  or 
Davis  quadrant;  the  sextant  of  Tycho,  which  was  also  used  by  the  Arabs  in 
the  tenth  century;  the  octants  of  Hooke  and  Fouchy,  in  which  a  mirror  was 
used  for  the  first  time;  and,  finally,  the  reflecting  octant  whose  principle  was 
due  to  Newton,  although  the  construction  was  first  carried  out  by  John  Hadley 
about  1731.  The  instrument  of  Hadley  has  been  improved  in  design,  but  no 
essential  modification  has  been  made  in  its  principle.  In  its  modern  form  it 
is  known  as  the  reflecting  sextant,  or  more  generally,  simply  as  the  sextant. 

With  the  exception  of  the  astrolabe  and  the  large  fixed  sextants  of  Tycho, 
the  various  forms  mentioned  are  characterized  by  the  fact  that  they  may  be 
held  in  the  hand  during  observations,  small  oscillations  and  variations  in  the 
position  of  the  instrument  offering  no  serious  difficulty  in  the  execution  of  the 
measures.  These  instruments  have  therefore  played  an  important  part  in  the 


86  PRACTICAL  ASTRONOMY 

practice  of  navigation,  and  to-day  the  sextant  is  the  only  instrument  which 
can  advantageously  be  employed  in  the  observations  necessary  for  the  deter- 
mination of  a  ship's  position.  In  addition,  its  compactness  and  lightness,  and 
the  precision  of  the  results  that  may  be  obtained  with  it  render  it  one  of  the 
most  convenient  and  valuable  instruments  at  our  command. 

The  modern  sextant  consists  of  a  light,  flat,  metal  frame  supporting  a 
graduated  arc,  usually  70°  in  length;  a  movable  index  arm;  two  small  mirrors 
perpendicular  to  the  plane  of  the  arc;  and  a  small  telescope.  The  index  arm 
is  pivoted  at  the  center  of  the  arc  and  has  rigidly  attached  to  it  one  of  the 
mirrors,  the  index  glass,  whose  reflecting  surface  contains  the  rotation  axis 
of  the  arm  and  the  attached  mirror.  The  position  of  the  index  glass  corre- 
sponding to  any  setting  may  be  read  from  the  graduated  arc  by  means  of  a 
vernier.  The  second  mirror,  the  horizon  glass,  is  firmly  attached  to  the 
frame  of  the  sextant  in  a  manner  such  that  when  the  vernier  reads  zero  the 
two  mirrors  are  parallel.  Only  that  half  of  the  horizon  glass  adjacent  to 
the  frame  is  silvered.  The  telescope,  whose  line  of  sight  is  parallel  to  the 
frame,  is  directed  toward  the  horizon  glass,  and  with  it  a  distant  object  may 
be  seen  through  the  unsilvered  portion.  When  the  frame  is  brought  into 
coincidence  with  the  plane  determined  by  the  object,  the  eye,  of  the  observer, 
and  any  other  object,  a  reflected  image  of  the  second  object  may  be  seen  in 
the  field  of  the  telescope,  simultaneously  with  the  first,  by  giving  the  index 
arm  a  certain  definite  position  depending  upon  the  angular  distance  separat- 
ing the  objects.  If  the  position  of  the  arm  is  such  that  the  rays  of  the  second 
object  reflected  by  the  index  glass  to  the  horizon  glass,  and  then  from  the 
silvered  portion  of  the  latter,  enter  the  telescope  parallel  to  the  rays  that  pass 
from  the  first  object  through  the  unsilvered  portion  of  the  horizon  glass,  the 
two  images  will  be  seen  in  coincidence.  This  being  the  case,  the  relative  incli- 
nation of  the  mirrors  as  shown  below,  will  be  one-half  the  angular  distance 
separating  the  objects;  and,  since  the  construction  is  such  that  the  inclination 
may  be  read  from  the  graduated  arc,  it  becomes  possible  to  find  the  angular 
distance  between  the  objects.  The  use  of  the  instrument  is  simplified  by 
graduating  the  arc  so  that  the  vernier  reading  is  twice  the  inclination  of  the 
mirrors,  and  hence,  directly,  the  angular  distance  of  the  objects.  With  the 
usual  form  of  the  instrument  the  maximum  angle  that  can  be  measured  is 
therefore  about  140°.  The  two  mirrors  and  the  telescope  are  provided  with 
adjusting  screws,  which  may  be  used  to  bring  them  accurately  into  the  posi- 
tions presupposed  by  the  theory  of  the  instrument.  In  addition,  the  tele- 
scope may  be  moved  perpendicularly  back  and  forth  with  respect  to  the  frame 
thus  permitting  an  equalization  of  the  intensity  of  the  direct  and  reflected 
images  by  varying  the  ratio  of  the  reflected  and  transmitted  light  that  enters 
the  telescope.  Adjustable  shade  glasses  adapt  the  instrument  for  observa- 
tions on  the  sun. 

49.  The  principle  of  the  sextant. — In  Fig.  10  let  0V  represent  the 
graduated  arc;  /and//,  the  index  glass  and  the  horizon  glass,  respectively; 
and  IV,  the  index  arm,  pivoted  at  the  center  of  the  arc  and  provided  with  a 


THE  SEXTANT  87 

vernier  at  V.  When  V  coincides  with  0,  the  mirrors  are  parallel.  The  posi- 
tion indicated  in  the  figure  is  such  that  the  two  objects  St  and  S2  are  seen  in 
coincidence,  for  the  rays  from  5r  pass  through  the  unsilvered  portion  of  H 
and  enter  the  telescope  in  the  direction  HE,  while  those  from  S2  falling  on  / 
are  reflected  to  //"and  thence  in  the  direction  HE.  The  two  beams  therefore 
enter  the  telescope  parallel. 


It  is  to  be  shown  that  the  inclination  of  7  to  H  is  one-half  the  angular 
distanced  separating  the  objects.  IN  and  HN  are  the  normals  to  the  mirrors, 
and  by  the  fundamental  laws  of  reflection  they  bisect  the  angles  S,IH  and 
IHE,  respectively.  In  the  triangle  IHE. 

2<z  =  2b  +  A, 
whence 

a  =  l>+  y2A. 
But  in  the  triangle  IHN 

a  =  b  +  M. 
Therefore, 

M=y2A. 

But  M,  being  the  angle  between  the  normals  to  the  mirrors,  measures  their 
inclination,  and  is  equal  to  the  angle  subtended  by  the  arc  0F,  whence 

A  =  20V.  (121} 

But  since  the  arc  is  graduated  so  that  the  reading  is  twice  the  angle  subtended 
by  OF  the  angular  distance  between  the  two  objects  is  given  directly  by  the 
scale. 

50.  Conditions  fulfilled  by  the  instrument. — The  following  conditions, 
among  others,  must  be  fulfilled  by  the  perfectly  adjusted  sextant. 


88  PRACTICAL  ASTRONOMY  . 

1.  The  index  glass  must  be  perpendicular  to  the  plane  of  the  arc. 

2.  The  horizon  glass  must  be  perpendicular  to  the  plane  of  the  arc. 

3.  The  axis  of  the  telescope  must  be  parallel  to  the  plane  of  the  arc. 

4.  The  vernier  must  read  zero  when  the  mirrors  are  parallel. 

5.  The  center  of  rotation  of  th£  index  arm  must  coincide  with  the  center 
of  the  graduated  arc. 

Since  the  positions  of  the  mirrors  and  the  telescope  are  liable  to  derange- 
ment, methods  must  be  available  for  adjusting  the  instrument  as  perfectly  as 
possible.  This  is  the  more  important  inasmuch  as  it  is  impossible  to  eliminate 
from  the  measures  the  influence  of  any  residual  errors  in  the  adjustments. 
Although  elimination  is  impossible,  it  should  be  remarked  that  the  errors 
arising  in  connection  with  Nos.  4  and  5,  at  least,  may  be  determined  by  the 
methods  given  in  Sections  52  and  53,  and  applied  as  corrections  to  the  read- 
ings obtained  with  the  instrument.  Conditions  Nos.  1-4  are  within  the  control 
of  the  observer.  No.  5  must  be  satisfied  as  perfectly  as  possible  by  the 
manufacturer. 

51.  Adjustments  of  the  sextant. — No.  i.  Index  glass.  To  test  the 
perpendicularity  of  the  index  glass,  place  the  sextant  in  a  horizontal  position, 
unscrew  the  telescope  and  stand  it  on  the  arc  just  in  front  of  the  surface  of 
the  index  glass  produced.  If  then  the  eye  be  placed  close  to  the  mirror,  the 
observer  will  see  the  reflected  image  of  the  upright  telescope  alongside  the 
telescope  itself.  By  carefully  moving  the  index  arm,  the  telescope  and  its 
image  may  be  brought  nearly  into  coincidence.  If  the  two  are  parallel,  the 
index  glass  is  in  adjustment.  The  telescope  should  be  rotated  about  its  axis 
in  order  to  be  sure  that  it  is  perpendicular  to  the  plane  of  the  arc.  If  the 
adjustment  is  imperfect,  correction  must  be  made  by  the  screws  at  the  base  of 
the  mirror.  Some  instruments  are  not  provided  with  the  necessary  screws, 
and  in  such  cases  the  adjustment  had  best  be  entrusted  to  an  instrument 
maker. 

The  test  can  also  be  made  by  looking  into  the  index  glass  as  before,  and 
noting  whether  the  arc  and  its  reflected  image  lie  in  the  same  place.  If  not, 
the  position  of  the  mirror  must  be  changed  until  such  is  the  case. 

No.  2.  The  horizon  glass.  The  adjustment  of  the  horizon  glass  may  be 
tested  by  directing  the  telescope  toward  a  distant,  sharply  defined  object, 
preferably  a  star,  and  bringing  the  index  arm  near  the  zero  of  the  scale.  Two 
images  of  the  object  will  then  be  seen  in  the  field  of  view — one  formed  by  the 
rays  transmitted  by  the  horizon  glass,  the  other,  by  those  reflected  into  the 
telescope  by  the  mirrors.  The  reflected  image  should  pass  through  the  direct 
image  as  the  index  arm  is  moved  back  and  forth  by  the  slow  motion.  If  it 
does  not,  the  horizon  glass  is  not  perpendicular  to  the  plane  of  the  arc,  and 
must  be  adjusted  until  the  direct  and  reflected  images  of  the  same  object 
can  be  made  accurately  coincident. 

No.  3.  The  telescope.  The  parallelism  of  the  telescope  to  the  frame  may 
be  tested  by  bringing  the  images  of  two  objects  about  120°  apart  into  coin- 
cidence at  the  edge  of  the  field  nearest  the  frame.  Then,  without  changing 


INDEX  CORRECTION  89 

the  reading,  shift  the  images  to  the  opposite  side  of  the  field.  If  they  remain 
in  coincidence,  the  telescope  is  in  adjustment.  If  not,  its  position  must  be 
varied  by  means  of  the  adjusting  screws  of  the  supporting  collar  until  the  test 
is  satisfactory. 

No.  4.  Index  adjustment.  If  the  fourth  condition  is  not  fulfilled,  an  index 
error  will  be  introduced  into  the  angles  read  from  the  scale.  To  test  the 
adjustment,  bring  the  direct  and  reflected  images  of  the  same  distant  object 
into  the  coincidence  as  in  the  adjustment  of  the  horizon  glass.  The  corres- 
ponding scale  reading  is  called  the  zero  reading  =  ^0.  If  R0  is  zero,  the 
adjustment  is  correct.  If  not,  set  the  index  at  o°,  and  bring  the  images  into 
coincidence  by  means  of  the  proper  adjusting  screws  attached  to  the  horizon 
glass.  //  is  better,  however,  to  disregard  this  adjustment  and  correct  the  readings 
by  the  amount  of  the  index  error. 

It  can  be  shown  that  the  errors  affecting  the  readings  as  a  result  of  an 
imperfect  adjustment  of  the  index  glass,  the  horizon  glass,  and  the  telescope 
are  of  the  order  of  the  squares  of  the  residual  errors  of  adjustment.  If  care  be 
exercised  in  making  the  adjustments,  the  resulting  errors  will  be  negligible 
as  compared  with  the  uncertainty  in  the  readings  arising  from  other  sources. 

52.  Determination  of  the  index  correction.  —  Make  a  series  of  zero 
readings  on  a  distant,  sharply  defined  object,  a  star  if  possible.  If  the  zero  of 
the  vernier  falls  to  the  right  of  the  zero  of  the  scale,  do  not  use  negative 
readings,  but  consider  the  last  degree  graduation  preceding  the  zero  of  the 
scale  as  359°,  and  read  in  the  direction  of  increasing  graduations.  The  zero 
reading  is  what  the  instrument  actually  reads  when  it  should  read  zero.  The 
index  correction,  /,.  is  the  quantity  which  must  be  added  algebraically  to  the 
scale  readings  to  obtain  the  true  reading.  We  therefore  have 

7=0°  —  *„,  (122) 

/=36o°  —  R0.  (123) 

The  latter  expression  is  to  be  used  for  the  determination  of  /  when  the 
zero  of  the  vernier  falls  to  the  right  of  that  of  the  scale  for  coincidence  of  the 
direct  and  reflected  images  of  the  same  object. 

When  observations  are  to  be  made  on  the  sun,  the  index  correction  should 
be  determined  from  measures  on  this  object.  Since  it  is  impossible,  on  account 
of  their  size,  to  bring  the  solar  images  accurately  into  coincidence,  we  deter- 
mine the  zero  reading  as  follows:  Make  the  two  images  externally  tangent, 
the  reflected  being  above  the  direct,  and  read  the  vernier.  Let  RI  represent 
the  mean  of  a  series  of  such  readings.  Then  make  an  equal  number  of  settings 
for  tangency  with  the  reflected  image  below.  Call  the  mean  of  the  corres- 
ponding readings  R2.  The  mean  of  Rt  and  R2  will  then  be  the  value  of  the 
zero  reading,  and  we  shall  have 


7=o°—  #(*,+*,),  (124) 

7=360°  —  j*(*x  +  /y.  (125) 


90  PRACTICAL  ASTRONOMY 

The  readings  thus  obtained  will  also  give  the  value  of  S,  the  sun's  semi- 
diameter.  Since  the  center  of  the  reflected  image  moves  over  a  distance  of 
four  semi-diameters  in  shifting  from  the  first  position  to  the  second,  we  have 

S=MR-R^.  (126) 

Owing  to  the  brilliancy  of  the  solar  image,  its  diameter  appears  larger 
than  it  really  is — a  phenomenon  known  as  irradiation.  Should  the  value  of 
5"  be  required  for  the  reduction  of  observations  on  the  sun  (see  Section  55), 
the  value  calculated  from  equation  (126)  should  be  used  rather  than  that 
derived  from  the  Ephemeris,  in  order  that  the  influence  of  irradiation  may  be 
eliminated. 

53.  Determination  of  eccentricity  corrections. — Any  defect  in  the 
fifth  condition  introduces  an  eccentricity  error  into  the  readings.  Since,  with 
the  usual  forjn  of  the  instrument  there  is  but  a  single  vernier,  this  cannot  be 
eliminated.  Each  sextant  must  be  investigated  specially  for  the  determination 
of  the  eccentricity  errors  affecting  the  readings  for  different  parts  of  the  scale. 
These  may  be  found  by  measuring  a  series  of  known  angles  of  different  mag- 
nitudes. The  mean  result  for  each  angle,  A,  gives  by  (71)  an  equation  of 
the  form 

A  =  R  +  I  +  E0-E<  (127) 

where  R  is  the  sextant  reading  for  coincidence  of  the  two  objects  whose 
angular  distance  is  A;  /,  the  index  correction;  and  E0  and  E,  the  eccentricity 
corrections  for  those  graduations  of  the  scale  which  coincide  with  vernier 
graduations  for  the  readings  RQ  and  R,  respectively.  The  readings  of  the 
coinciding  graduations  when  the  vernier  reads  R0  and  R  may  be  denoted  by 
R0'  and  R',  respectively.  E0  —  E  is  the  correction  which  must  be  applied  to 
the  sextant  reading,  freed  from  index  correction,  in  order  to  obtain  the  true 
value  of  the  angle.  Denoting  its  value  by  e,  (127)  may  be  written 

e=A  —  (R  +  I).  (128) 

Having  determined  e  from  (128)  for  a  considerable  number  of  angles  dis- 
tributed as  uniformly  as  possible  over  the  scale,  the  results  may  be  plotted  as 
ordinates  with  the  corresponding  values  of  R'  as  abscissas.  From  the  plot  a 
table  may  be  constructed  giving  the  values  of  £  for  equidistant  values  of  R', 
from  which  the  value  of  e  for  any  other  reading,  R,  can  then  be  derived.  Care 
should  be  taken  always  to  enter  the  table  with  the  R'  corresponding  to  the 
given  R  as  argument.  It  should  be  noted  that  the  usefulness  of  the  table 
depends  upon  /  remaining  sensibly  constant,  for  if  the  index  correction 
changes  by  any  considerable  amount,  R0'  may  change  sufficiently  to  render 
the  tabular  values  of  £  no  longer  applicable. 

The  chief  difficulty  in  investigating  the  eccentricity  of  a  sextant  consists 
in  securing  a  suitable  series  of  known  angles.  A  simple  method  is  to  measure 
with  a  good  theodolite  the  angles  between  a  series  of  distant  objects,  nearly 


MEASUREMENT  OF  ALTITUDES  91 

in  the  horizon,  care  being  taken  to  tilt  the  instrument  so  that  in  turning  from 
one  object  to  the  next  no  rotation  about  the  horizontal  axis  is  necessary. 

54.  Precepts  for  the  use  of  the  sextant. — The  following  points  should 
carefully   be    noted    in    using    the    sextant:     Focus  the  telescope  accurately. 
The  image  of  a  star  should  be  a  sharply  defined  point;  that  of  the  sun  must 
show  the  limb  clearly  defined  and  free  from  all  blurring.     For  solar  observa- 
tions, use,  whenever  possible,  shade  glasses  attached  to  the  eyepiece  rather 
than  those  in  front  of  the  mirrors;  and  reduce  the  intensity  of  the  images  as 
much  as  is  consistent  with  clear  definition.     If  the  use  of  the  mirror  shade 
glasses    cannot   be  avoided,  select    those   which    will    make    the   direct   and 
reflected   images  of  the  same  color,  and   reverse  them  through    180°  at  the 
middle  of  the  observing  program  to  eliminate  the  effect  of  any  non-parallelism 
of  their  surfaces.     If  a  roof  is  used  to  protect  the  surface  of  the  mercury  from 
wind,  it  also  should  be  reversed  at  the  middle  of  the  program.     In  all  cases 
make  the  direct  and  reflected  images  of  the  same  intensity  by  regulating  the 
distance  of  the  telescope  from  the  frame.     Make  the  adjustments  in  the  order 
in  which  they  are  given  above,  and  always  test  them  before  beginning  obser- 
vations.    The  index  correction  should  be  determined  both  before   and  after 
each  series  of  settings.     Make  all  coincidences  and  contacts  in  the  center  of 
the  field.     Finally,  the  instrument  should  be  handled  with  great  care,  for  a 
slight  shock  may  disturb  the  adjustment  of  the  mirrors  and  change  the  value 
of  the  index  correction. 

55.  The   measurement   of  altitudes. — Although  the  sextant  may  be 
used  for  the  measurement  of  angles  lying  in  any  plane,  it  finds  its  widest 
application  in  practical   astronomy  in  the  determination  of  the  altitude  of  a 
celestial  body. 

At  sea  the  observations  are  made  by  bringing  the  reflected  image  of  the 
body  into  contact  with  the  image  of  the  distant  horizon  seen  directly  through 
the  unsilvered  portion  of  the  horizon  glass.  To  obtain  the  true  reading  the 
plane  of  the  arc  must  be  vertical.  Practically,  the  matter  is  accomplished  by 
rotating  the  instrument  back  and  forth  slightly  about  the  axis  of  the  tele- 
scope, which  causes  the  reflected  image  to  oscillate  along  a  circular  arc  in  the 
field.  The  index  is  to  be  set  so  that  the  arc  is  tangent  to  the  image  of  the 
horizon.  The  corresponding  reading  corrected  for  index  correction,  dip  of 
horizon,  and  refraction  is  the  required  altitude.  The  correction  for  dip  is 
necessary,  since,  owing  to  the  elevation  of  the  observer,  the  visible  horizon 
lies  below  the  astronomical  horizon.  The  square  root  of  the  altitude  of  the 
observer  above  the  level  of  the  sea,  expressed  in  feet,  will  be  the  numerical 
value  of  the  correction  in  minutes  of  arc.  The  observations  are  not  suscept- 
ible of  high  precision,  and  the  correction  for  eccentricity  may  be  disregarded 
as  relatively  unimportant. 

For  observation's  on  land  the  artificial  horizon  must  be  used.  The  meas- 
urement of  the  angular  distance  between  the  object  and  its  mercury  image 
gives  the  value  of  the  double  altitude  of  the  object.  Some  practice  is 
required  in  order  to  be  able  to  bring  the  object  and  its  mercury  image  into 


92 

coincidence  quickly  and  accurately.  In  case  the  object  is  a  star,  care  must  be 
taken  that  the  images  coinciding  are  really  those  of  the  object  and  its  reflec- 
tion in  the  mercury.  The  following  is  the  simplest  method  of  procedure: 
Stand  in  a  position  such  that  the  mercury  image  is  clearly  visible  in  the 
center  of  the  horizon,  and  direct  the  telescope  toward  the  object.  By  bring- 
ing the  index  near  zero  the  reflected  image  will  appear  in  the  field.  The 
telescope  is  then  turned  slowly  downward  toward  the  mercury,  the  index 
being  moved  forward  along  the  arc  at  the  same  time  at  a  rate  such  that  the 
reflected  image  of  the  object  remains  constantly  in  the  field.  If  the  plane  of 
the  sextant  is  kept  vertical,  and  if  the  observer  is  careful  to  stand  so  that  the 
mercury  reflection  can  be  seen,  its  image  seen  directly  through  the  unsilvered 
portion  of  the  horizon  glass  will  come  into  the  field  when,  the  telescope  has 
been  sufficiently  lowered.  Both  images  should  then  be  visible.  The  varying 
altitude  of  the  object  will  cause  them  to  change  their  relative  positions.  The 
index  is  set  so  that  the  images  are  approaching  and  clamped.  When  they 
become  coincident  the  time  is  noted  and  the  vernier  read.  The  instant  of 
coincidence  is  best  determined  by  giving  the  instrument  a  slight  oscillatory 
motion  about  the  axis  of  the  telescope  and  noting  the  time  when  the  reflected 
image  in  its  motion  back  and  forth  across  the  field  passes  through  the  direct 
image. 

To  obtain  an  accurate  value  of  the  altitude,  a  series  of  such  settings 
should  be  taken  in  quick  succession,  the  time  and  the  vernier  reading  being 
noted  for  each.  It  is  not  necessary  to  use  the  method  described  above  for 
bringing  the  images  into  the  field  for  any  of  the  settings  but  the  first;  for  if, 
after  reading,  the  index  be  left  clamped  and  the  telescope  be  directed  toward 
the  mercury  image,  the  plane  of  the  arc  being  held  vertical,  the  reflected 
image  will  also  be  in  the  field.  If  it  is  not  at  once  seen,  a  slight  rotation 
about  the  axis  of  the  telescope  will  bring  it  into  view,  unless  too  long  an 
interval  has  elapsed. 

Measures  for  altitude  may  also  be  made  by  setting  the  zero  of  the  vernier 
accurately  on  one  of  the  scale  divisions  so  that  the  images  are  near  each  other 
and  approaching  a  coincidence.  The  time  of  coincidence  and  the  vernier 
reading  are  noted.  The  index  is  then  moved  20'  so  that  the  images  will  again 
be  approaching  coincidence.  The  time  and  the  reading  are  noted  as  before 
and  the  process  is  repeated  until  a  sufficient  number  of  measures  has  been 
secured. 

The  consistency  of  the  measures  should  always  be  tested,  as  in  the  case 
of  the  engineer's  transit  (see  page  79)  by  calculating  the  rate  of  change  of  the 
readings  per  minute  of  time.  If  however,  the  observations  have  been  made 
by  noting  the  times  of  coincidence  for  equidistant  readings  of  the  vernier,  the 
constancy  of  the  time  intervals  between  the  successive  settings  will  be  a 
sufficient  test. 

If  R  denote  the  mean  of  the  vernier  readings,  the  apparent  double  altitude 
of  the  object  will  be  given  by 

(129) 


MEASUREMENT  OF  ALTITUDES  93 

in  which  /is  the   index  correction,  and  e  the  correction  for  eccentricity.     The 
true  altitude  corresponding  to  the  mean  of  the  observed  times  is  found  from 


where  the  refraction,  r,  may  be  derived  from  Table  I,  page  20,  or  if  more 
accurate  results  are  required,  by  equation  (3),  page  18.  If  the  zenith  distance 
is  desired  instead  of  the  altitude,  we  calculate  z'  from 

z'  =  go°—h',  (130) 

and  z  from 

z  =  z'  +  r.  (131) 

For  measures  on  the  sun  coincidences  are  not  observed,  but,  instead,  the 
instants  when  the  images  are  externally  tangent.  To  eliminate  the  influence 
of  semidiameter,  the  same  number  of  contacts  should  be  observed  for  both 
images  approaching  and  images  receding.  If  for  any  reason  this  cannot  be 
done,  a  correction  for  semidiameter  must  be  applied.  Let 

«a  =  number  of  settings  for  images  approaching, 

wr  =  number  of  settings  for  images  receding, 

n  =  total  number  of  settings, 

5  =  the  semidiameter  of  the  sun  calculated  by  equation  (126). 

We  shall  then  have  for  solar  observations 

?/.'—  p-u^a  —  nr  g.   .     ,  /Upper  sign,  altitude  decreasing.)  (        ^ 

n  £'    \  Lower  sign,  altitude  increasing.  / 

in  which  h'  is  the  apparent  altitude  of  the  sun's  center  corresponding  to  the 
mean  of  the  observed  times;  and  the  term  involving  S,  the  correction  for 
semidiameter.  The  true  altitude  and  zenith  distance  are  then  given  by 


(133) 
z  —  z'  +  r—p,  (134) 

The  solar  parallax,/,  may  be  obtained  from  columns  four  an-d  eight  of  Table 
I,  page  20.  For  approximate  results  r  —  p  may  be  taken  from  the  fifth 
and  tenth  columns  of  this  same  table. 

Example  85.  On  1909,  April  10,  the  following  sextant  observations  of  the  altitude  of 
the  sun  were  made  at  the  Laws  Observatory  near  the  time  of  meridian  transit.  The  error  of 
the  timepiece  was  A0f  =  +  6m  37s.  The  observations  will  be  reduced  later  for  the  determina- 
tion of  latitude. 


94 


PRACTICAL  ASTRONOMY 


Readings  on  Sun 


for 

Index  Correction                        0F 

Reading 

Limb 

/? 

R* 

oh 

59H 

1  22s 

117° 

18' 

50" 

Lower 

0° 

31' 

30" 

359°  28' 

0"              I 

i 

28 

118 

23 

10 

Upper 

3i 

40 

28 

20 

2 

29 

118 

25 

50 

Upper 

32 

0 

28 

o     •" 

4 

9 

117 

24 

o 

Lower 

32 

10 

28 

10 

5 

19 

117 

24 

30 

Lower 

0 

,r 

5° 

359     28 

8 

6 

53 

118 

28 

40 

Upper 

8 

27 

118 

29 

o 

Upper 

ro  Reading 

=  359     59 

59 

12 

8 

117 

24 

IO 

Lower 

Index  Corr.        =  -j-  i 

/?!  —  J?2=      i       3     42 

Semidiameter   =  15     56 


CHAPTER  V 

THE  DETERMINATION  OF  LATITUDE 

56.  Methods. — On  page  34  it  was  shown  that  if  the  zenith  distance  or 
altitude  of  a  star  of  known  right  ascension  and  declination  be  measured  at  a 
known  time,  the  latitude  of  the  place  of  observation  can  be  determined  by 
means  of  equation  (31).  The  preceding  chapter  indicates  the  methods  that 
may  be  employed  for  the  measurement  of  the  zenith  distance.  It  is  the  pur- 
pose of  the  present  chapter  to  determine  the  most  advantageous  method  of 
using  the  fundamental  equation  and  to  develop  the  formulae  necessary  for 
the  practical  solution  of  the  problem. 

To  establish  a  criterion  for  the  use  of  equation  (31),  it  is  to  be  noted  that 
the  resultant  error  of  observation  in  <p  will  depend  upon  the  errors  affecting 
o,  z,  d,  and  a.  Star  positions  are  so  accurately  known,  however,  that  the 
errors  in  a  and  d  are  insignificant  as  compared  with  those  occurring  in  z  and 
6;  and  we  need  concern  ourselves  only  with  those  affecting  the  latter  two 
quantities.  It  is  particularly  important  to  know  the  influence  of  an  error  in 
the  time,  for  since  this  quantity  is  assumed  to  be  known,  it  is  desirable  to  be 
able  to  specify  how  accurately  it  must  be  given  in  order  to  obtain  a  definite 
degree  of  precision  in  the  latitude. 

The  relation  connecting  small  variations  in  z  and  6  with  changes  in  <p  is 
found  by  differentiating  (31),  z,  t=d  —  a,  and  <p  being  considered  variable. 
(Num.  Comp.  p.  II.)  We  thus  find 

—  sin  zdz  =  sin  d  cosy  dtp  —  cos  d  sin  <p  cos/  dtp  —  cos  d  cos  y  sin  tdt,      (135) 
which  by  means  of  (32)  and  (33)  reduces  to 

dz  =  cos  A  d(p  -\-  sin  A  cos  <p  dt. 
Writing  dt=dd  and  solving  for  dip 

dy  =  se.cAdz —  tan  A  cos  <pdd.  (J36) 

Assuming  now  that  the  differentials  of  z  and  d  represent  the  errors  in 
these  quantities,  the  resultant  error  in  <p  will  be  given  by  (136).  In  order  that 
this  may  be  a  minimum,  sec  A  and  tan  A  must  have  their  minimum  absolute 
values,  which  will  occur  when  A  is  o°  or  180°.  Since  these  quantities  increase 
as  the  azimuth  deviates  from  o°  or  180°,  the  object  observed  for  the  deter- 
mination of  latitude  should  be  as  near  the  meridian  as  possible.  Even  with 
this  limitation  there  will  be  considerable  variety  in  the  procedure  depending 
upon  the  position  of  the  star  and  the  circumstances  of  the  observations;  and 
we  now  proceed  to  the  consideration  of  the  following  five  cases  in  which  the 
given  data  are,  respectively, 

95 


96  PRACTICAL  ASTRONOMY 

1.  The  zenith  distance  of  an  object  when  on  the  meridian, 

2.  The  difference  of  the  meridian  zenith  distances  of  two  stars, 

3.  A  series  of  zenith  distances  when  the  object  is  near  the  meridian, 

4.  The  zenith  distance  of  an  object  at  any  hour  angle, 

5.  The  altitude  of  Polaris  at  any  hour  angle. 

I.       MERIDIAN    ZENITH    DISTANCE 

57.     Theory.  —  The  hour  angle  of  an  object  on  the  meridian  is  zero.     For 
this  case  equation  (31)  reduces  to 

cosz0  =  cos((f>  —  <?),  (137) 

whence 

or 


Equation  (138)  may  also  be  derived  geometrically  by  means  of  Fig.  4,  p.  24 
whence  it  is  seen  that  the   upper  sign  must  be   used   for  objects  south  of  the 
zenith;  and    the  lower,  for  objects  between   the  zenith    and   the  pole.     For 
lower  culmination  the  fundamental  relation  becomes 

^  =  180°  —  d  —  s0.  (139) 

58.  Procedure.  —  For  the  instant  of  observation  we  have  by  (35)  6  =  a. 
If  Ad  be  the  error  of  the  timepiece,  the  clock  time  of  transit  will  be 

6'  =  a  —  J0,  (140) 

where  a,  along  with  <?,  is  to  be  interpolated  from  the  Ephemeris  for  the  instant 
of  observation.  The  true  zenith  distance  is  then  to  be  determined  by  some 
one  of  the  methods  of  Section  45  or  55  for  the  clock  time  d'.  Equation  (138) 
or  (139)  will  then  give  the  required  value  of  the  latitude. 

If  a  mean  timepiece  is  used,  the  sidereal  time  of  transit  must  be  converted 
into  the  corresponding  mean  time,  7",  by  equations  (62)  and  (41),  pp.  49  and  39, 
respectively.  The  clock  time  of  obervation  is  then  given  by 

T'=T—JT.  (141) 

In  case  the  error  of  the  timepiece  is  uncertain,  the  observer  will  bring  the 
image  to  the  intersection  of  the  threads,  or  the  direct  and  reflected  images 
into  coincidence  if  the  sextant  is  used,  a  little  before  the  time  of  transit  and 
follow  with  the  slow  motion  until  it  becomes  necessary  to  reverse  the  direc- 
tion in  which  the  tangent  screw  is  turned  in  order  to  keep  the  image  on  the 
thread.  This  instant  marks  the  time  of  meridian  passage.  The  corresponding 
reading,  properly  corrected,  then  gives  the  altitude  as  before. 

Example  36.  On  1909,  April  10,  an  observation  was  made  at  the  Laws  Observatory  with 
a  sextant  for  the  determination  of  the  latitude  by  a  meridian  altitude  of  the  sun.  The  reading 
on  the  upper  limb  at  the  calculated  time  of  transit  was  118°  29'  10".  The  error  of  the  clock, 


TALCOTT'S  METHOD  97 

the  index  correction,  and  the  semidiameter  to  be  used  are  those  of  Ex.  35.  The  calculation 
of  the  clock  time  of  transit  is  in  the  left  hand  column.  The  reduction  of  the  observation  for 
the  determination  of  the  latitude  is  in  the  second  column. 


Gr.  A.  T.  of  Col.  A.  N.  —  6h    9mi8"  =  6Si55  R   =118°  29'  10" 

Sun's  a  at  Col.  A.  N.  =  i     14     42  I    =  -f-  i 

J0F=    +  6     37  £     :  Unknown 

0'  =  i       8       5  2h'  =  118     29     ii 

V  =    59  H  36 

The  true  value  of  the  latitude  is                                  z'  =    30  45  24 

known  to  be  38°  56'  152"  r  —  p='  30 

^  =  15  56 

z  —    31  i  50 

8  =+7  54  29 

<p  =   38  56  19 

2.     DIFFERENCE  OF  MERIDIAN  ZENITH  DISTANCES 
TALCOTT'S   METHOD 

59.     Theory.  —  From  equation  (138)  we  have 

<  =  ds  +  zs  , 


where  the  subscripts  indicate  the  position  of  the  stars  with  respect  to  the 
zenith.  One-half  the  sum  of  these  two  equations  gives 

V  =  &  (8S  +  8»)  +  tf(zs'  —  Zx')+I/2(rs  —  rM),  (142) 

in  which  the  true  zenith  distances  have  been  replaced  by  #/  +  r^  and  zs'  +  r&, 
respectively.  The  declinations  are  given  by  the  Ephemeris^  and  the  difference 
of  the  refractions  is  readily  calculated.  If  therefore  the  difference  between 
the  apparent  zenith  distances  of  two  stars  be  measured,  the  latitude  can  be 
calculated  by  (142). 

By  limiting  the  application  of  the  equation  to  those  cases  in  which  the 
zenith  distances  are  nearly  equal,  a  considerable  increase  in  precision  will  be 
obtained  as  compared  with  that  resulting  from  meridian  zenith  distances. 
Since  the  measures  are  differential,  instrumental  errors  affecting  the  two  ob- 
servations equally  will  be  eliminated.  In  the  case  of  measures  with  the 
sextant,  for  example,  the  index  correction  and  the  eccentricity  will  be  elim- 
inated and  need  not,  therefore,  be  determined.  But  what  is  of  more  import- 
ance, so  far  as  precision  is  concerned,  is  the  fact  that  the  errors  of  observation 
which  would  affect  these  instrumental  corrections,  were  they  determined,  do 
not  enter  into  the  result.  A  similar  condition  exists  in  the  case  of  the  refrac- 
tion, for  the  difference  of  two  refractions  corresponding  to  nearly  equal  zenith 
distances  can  be  calculated  with  a  higher  degree  of  precision  than«is  possible 
in  the  determination  of  the  total  refraction.  Finally,  the  fact  that  the  quantity 
to  be  observed  is  small,  makes  it  possible  to  introduce  other  and  more  precise 
methods  of  measurement  than  those  which  depend  upon  the  use  of  a  grad- 
uated circle.  For  example,  with  the  engineer's  transit  small  differences  of 
7 


98 


PRACTICAL  ASTRONOMY 


zenith  distance  may  be  measured  more  accurately  with  the  gradienter  screw 
than  with  the  vertical  circle. 

The  method  under  discussion  was  first  proposed  by  Horrebow,  the  director 
of  the  Observatory  of  Copenhagen  about  the  middle  of  the  eighteenth  century, 
and  was  given  extensive  practical  application  in  the  work  of  the  United  States 
Coast  and  Geodetic  Survey  about  a  century  later  by  Captain  Talcott,  from 
which  circumstance  it  is  commonly  known  as  Talcott's  method.  It  reaches  its 
highest  precision  when  used  in  connection  with  the  zenith  telescope,  an  in- 
strument of  the  altazimuth  type  fitted  with  an  accurately  constructed  micro- 
meter eyepiece  and  a  very  sensitive  altitude  level.  The  level  enables  the 
observer  to  give  the  line  of  sight  the  same  inclination  to  the  vertical  during 
both  observations,  while  the  micrometer  affords  a  very  precise  determination 
of  the  required  difference  in  zenith  distance  of  the  two  stars. 

If  the  method  is  to  be  used  in  connection  with  the  engineer's  transit,  the 
angular  value  of  one  revolution  of  the  gradienter  screw  should  first  be  deter- 
mined by  measuring  a  small  angle  whose  value  is  known.  The  observa- 
tions should  be  made  and  reduced  in  a  way  such  that  any  irregularity  in  the 
screw  will  be  revealed.  To  this  end  a  process  analogous  to  that  used  in 
Examples  26  and  29  may  be  employed. 

Since  the  correction  for  refraction  will  always  be  small,  we  may  assume 


From  (4)  we  find 


dr 


dr  ,   . 

-T-  =  57   sec2 z  sin  i  , 
dz 


which  expresses  the  rate  of  change  of  r  per  i°  of  change  in  z'.  Denoting  this 
quantity  by  C,  the  correction  for  refraction  in  seconds  of  arc  becomes 

y2(r — rtl)"=%(zs'  —  Zn'}°C  (T43) 

in  which  the  difference  of  the  ze-nith  distances  must  be  expressed  in  degrees. 
The  value  of  C  may  be  taken  from  Table  V  with  the  mean  zenith  distance  of 
the  two  stars  as  argument. 

TABLE  V 


z' 

C 

*' 

C 

10° 

x'.'o 

50° 

2  '.'4 

20 

I.I 

55 

3-o 

30 

I.J 

60 

4.0 

40 

i-7 

65 

5-6 

50 

2-4 

70 

8-5 

60.     Procedure. — Select  two  stars  culminating  within  I5m  or  20m  of  each 
other  whose  declinations  satisfy  as  nearly  as  possible  the  condition 


CIRCUMMERIDIAN  ALTITUDES  99 

and  calculate  the  clock  time  of  meridian  transit  by  (140)  or  (141). 

If  a  sextant  is  used,  measure  the  double  altitudes  of  the  two  stars  at  the 
instants  of  transit.  Let  ^s  and  R*  be  the  corresponding  sextant  readings. 
The  second  term  of  (142)  will  then  be  given  by 

#(*.'  — *«')=#(*«  —  *.).  (145) 

If  the  engineer's  transit  is  employed,  level  carefully  and  bring  the  star 
culminating  first  to  the  intersection  of  the  threads  at  the  instant  of  its  transit. 
Read  the  gradienter  screw,  reverse,  relevel,  bring  the  second  star  to  the  inter- 
section of  the  threads  at  the  instant  of  transit  by  means  of  the  screw,  and  note 
the  reading  as  before.  The  vertical  circle  should  be  firmly  clamped  when 
the  setting  on  the  first  star  is  made,  and  must  not  be  disturbed  thereafter 
until  the  second  star  has  been  observed.  If  the  two  screw  readings  be 
denoted  by  ms  and  m^  and  if  G  be  the  value  of  one-half  a  revolution  of  the 
screw,  we  shall  have 

yz(z\  —  z'v)  =  ±G(ms  —  ;»„),  (146) 

in  which  the  upper  sign  is  to  be  used  when   the  screw  readings  increase  with 
increasing  zenith  distance. 

In  levelling,  special  attention  should  be  given  to  the  altitude  level. 
Unless  the  bubble  has  the  same  position  for  both  observations,  an  error  will 
be  introduced  into  the  result.  If  the  level  is  a  sensitive  one,  it  will  be  better 
to  omit  the  levelling  after  reversal  and  apply  a  correction  to  the  result  given 
by  (146).  If  o  and  e  be  the  readings  of  the  object  and  eye  ends  of  the 
bubble,  respectively,  and  if  readings  increasing  toward  the  north  be  recorded 
as  positive  while  those  increasing  toward  the  south  are  entered  as  negative, 
the  correction  to  be  added  algebraically  to  the  result  given  by  (146)  will  be 

(p*+et+o*+e^D,  (147) 

in  which  D  is  one-fourth  the  angular  value  of  one  division  of  the  level.     The 
bubble  readings  should  be  taken  as  near  the  times  of  transit  as  possible. 

The  last  term  of  (142)  is  given  by  (143),  the  value  of  C  being  derived  from 
Table  V.  The  declinations  are  to  be  taken  from  the  list  of  apparent  places  in 
the  Ephemeris  for  the  instant  of  observation.  In  case  the  northern  star  is  ob- 
served at  lower  culmination,  its  declination  in  (142)  must  be  replaced  by 
180°  — 0N. 

3.      CIRCUMMERIDIAN    ALTITUDES 

61.  Theory. — The  zenith  distance  to  be  used  in  equations  (138)  and  (139) 
is  that  of  the  object  when  on  the  meridian.  Since  only  a  single  determination 
of  this  quantity  can  be  made  at  any  given  transit,  it  is  desirable  for  the  sake  of 
precision  to  modify  the  method  described  under  No.  I  so  as  to  permit  a  multi- 
plication of  the  settings. 

The  change  in  the  zenith  distance  during  an  interval  immediately  preced- 
ing or  following  the  instant  of  transit  is  small  and  its  value  is  easily  and 


100  PRACTICAL  ASTRONOMY 

accurately  calculated.  The  meridian  zenith  distance  may  therefore  be  found 
by  observing  when  the  object  is  near  the  meridian  and  applying  to  the  meas- 
ured value  of  the  coordinate  the  amount  of  the  change  during  the  interval 
separating  the  instant  of  observation  from  that  of  culmination.  A  series  of 
such  measures  reduced  to  the  meridian  gives  a  precise  value  of  s0  which  can 
then  be  substituted  into  (138)  or  (139)  for  the  determination  of  the  latitude. 
It  is  of  course  immaterial  whether  the  quantity  measured  be  zenith  distance  or 
altitude.  The  method  is  commonly  known  as  that  of  circummeridian  altitudes. 
The  development  of  the  formulae  to  be  used  for  the  calculation  of  the 
reduction  to  the  meridian  is  as  follows:  Equation  (31)  may  be  written  in  the 
form 

cos  2  =  cos  (<p  —  d)  —  2  cos  <p  cos  d  sin7  l/2  t.  (*48) 

Let  z  be  the  observed  value  of  the  coordinate,  z0  the  meridian  value,  and  Zthe 
reduction  to  the  meridian.  We  then  have 

z  +  Z=z0.  (149) 

Substituting  into  (148)  we  find 

cos  (z0  —  Z]  =cos  z0  —  2  cos  <f>  cos<5  sin2  YT.  t.  (150) 

To  express  Z  explicitly  we  may  replace  the  left  member  of  (150)  by  its  expan- 
sion by  Taylor's  theorem.  Since  Z  is  small  the  convergence  will  be  rapid. 
Introducing  at  the  same  time 

A  =  cos  <p  cos  d  cosec  #0,  m  =  2  sin*1/^?,  (151) 

and  neglecting  terms  in  Z3  we  find 

Z=  —  Am+y2Z*cotz0.  (152) 

Squaring,  we  have  to  the  same  degree  of  approximation 

Z2  =  A2m*. 
Substituting  into  (152),  and  writing 

£=A'cotz0,  n=y2m*=2sin4l/2t,  (153) 

we  have  finally  for  the  reduction  to  the  meridian. 

(154) 


Since  the  observations  may  be  arranged  so  that  Z  will  not  exceed  15'  or  20', 
the  error  in  (154)  will  be  insensible. 

Combining  equations  (138),  (149),  and  (154),  the  expression  for  the  latitude 
becomes 

Bn,  (155) 


CIRCUMMERIDIA  N  AL  T ITU  DBS 


101 


in  which  the  upper  sign  is  to  be  used  for  southern  stars;  and   the  lower,  for 
those  culminating  between  the  zenith  and  the  pole. 

For  an  object  observed  near  lower  culmination,  /  in  (31)  must  be  replaced 
by  180°  -|-  t.  The  resulting  value  of  the  reduction  to  the  meridian  substituted 
into  (139)  gives 


=  180°  —  3  —  z  —  Am  —  Bn. 


(156) 


Equations  (155)  and  (156),  in  which  the  last  terms  are  to  be  calculated  by 
(151)  and  (153),  express  the  solution  of  the  problem.  For  observations  with 
the  engineer's  transit  the  term  Bn  will  usually  be  insensible  when  the  hour 
angle  is  less  than  I5m  or  2Om. 

It  will  be  observed  that  A  and  B  depend  upon  the  latitude  —  the  quantity 
to  be  determined.  A  value  of  <p  sufficiently  accurate  for  the  calculation  of 
these  coefficients  may  be  obtained  by  (138)  or  (139)  from  the  value  of  z 
observed  nearest  the  time  of  transit.  It  will  be  noted  further  that  A  and  B  are 
constant  for  any  given  series  of  observations  and  need  be  calculated  but  once. 
The  factors  m  and  «,  on  the  other  hand,  are  different  for  each  setting.  Since 
they  depend  only  upon  the  hour  angle,  their  values  may  be  tabulated  with  /  as 
argument.  Tables  VI  and  VII  may  be  used  for  all  ordinary  observations  with 
the  transit  or  sextant. 


TABLE  VI 


TABLE  VII 


t 

;« 

t 

M 

Om 

o" 

I0m 

196" 

2 

42 

I 

2 

ii 

2  ^8 

2 

8  6 

12 

*S34S 

IO 

49 

3 

18 

13 

332 

4 

13 
31  18 

H 

*  I 

5 

49  22 

15 

442  , 
60 

6 

7 

,  25 
9 

16 

502 

,  65 

57  6 

8 

126 

18 

636 

9 

33 
159 

19 

72 
708  ' 

10 

,  37 
196 

20 

77 
785   . 

t 

H 

Om 

o'.'o 

5 

O.O 

10 

O.I 

1.5 

o-5 

16 

0.6 

17 

0.8 

18 

I.O 

19 

I  .2 

20 

i  5 

62.  Procedure. — Calculate  trre  clock  time  of  transit,  6'  or  T',  of  the 
object  to  be  observed.  Beginning  a  few  minutes  before  this  instant,  make  a 
series  of  observations  for  the  determination  of  the  zenith  distance  by  some 
one  of  the  methods  of  Section  45  or  55,  noting  the  time  for  each.  Correct  the 
apparent  zenith  distance  for  refraction  in  the  case  of  a  star,  and  for  refraction 
and  parallax  in  the  case  of  the  sun.  Form  the  hour  angle,  /,  corresponding  to 
each  observation  by  subtracting  the  clock  time  of  transit  from  the  time  of 
observation.  For  a  star,  /  must  be  expressed  in  sidereal  units;  for  the  sun,  in 
solar  units.  Calculate  A  and  B  from  (151)  and  (153),  using  the  value  of  the 


102  PRACTICAL  ASTRONOMY 

zenith  distance  observed  nearest  the  time  of  transit  for  s0  and  for  the  determi- 
nation of  an  approximate  value  of  ^>,  both  of  which  are  required  for  the  com- 
putation. Finally,  calculate  the  latitude  for  each  observation  by  means  of 
(155)  or  (156).  The  declination  to  be  used  is  that  corresponding  to  the  instant 
of  observation. 

The  final  result  may  also  be  obtained  by  applying  the  mean  of  all  the 
values  of  Am  and  of  Bn  to  the  mean  of  all  the  zenith  distances  in  accordance 
with  equations  (155)  or  (156).  This  method,  however,  gives  no  indication  as 
to  the  consistency  of  the  observations,  and  it  is  better  to  reduce  the  results 
separately,  or,  at  least,  to  reduce  separately  the  means  of  not  more  than  two 
or  three  consecutive  measures. 

The  method  of  circummeridian  altitudes  may  advantageously  be  com- 
bined with  that  of  Talcott.  When  this  is  done  there  will  be  given  a  series  of 
values  of  ^  (z\ — £'N)  derived  from  observations  made  near  the  meridian. 
Each  of  these  must  be  reduced  to  the  meridian  by  adding  to  (142)  the  term 
y-2  (Zs- — ZN),  in  which  Zs  and  ZN  are  to  be  calculated  by  (154). 

Example  37.  The  reduction  of  the  circummeridian  altitudes  given  in  Ex.  35,  p.  93,  is  as 
follows:  To  eliminate  the  semidiameter  the  means  are  formed  for  the  ist  and  2nd,  3rd  and 
4th,  5th  and  6th,  and  the  7th  and  8th  observations.  These  results  are  in  the  first  and  sixth 
lines  of  the  calculation  below.  The  eccentricity  corrections  are  unknown.  The  index  cor- 
rection found  in  Ex.  35  is  -f-  i".  In  Ex.  36,  p.  96,  the  clock  time  of  transit  was  found  to  be 
ih  8m  5s. 


/  (sidereal) 
t  (solar) 
m 

jh   om  2f* 

—7  40 
—7  39 
115" 

—4 
—4 

'  19" 
46 

45 
44" 

jh   fcm 
—  I 
—  I 

6» 

59 
59 

8" 

Ih  lOm 

+  2 
+  2 

18" 

13 
10" 

n 

0 

o 

0 

0 

R 

ii7c 

'  Si' 

o" 

"7°  54' 

55" 

117° 

56' 

35" 

II70  56' 

35" 

h' 

58 

55 

30 

58  57 

28 

58 

58 

18 

58   58 

18 

z' 

3i 

4 

30 

31    2 

3i 

3i 

i 

42 

31    I 

42 

r-p 

+30 

+30 

+30          -I 

-3° 

3 

+7 

54 

22 

+7  54 

25 

+7 

54 

27 

+  7  54 

3i 

—  Am 

2 

52 

—  i 

6 

—  0 

12 

—0 

15 

9 

38 

56 

30 

38  56 

20 

38 

56 

27 

38  56 

28 

From  the  3rd 

column 

COS  £> 

9- 

8908 

P 

38° 

57' 

COS(J 

9- 

9959 

3 

+7 

54 

cosec  z^ 

>  °- 

2877 

zo 

31 

2 

log  A 

o. 

1744 

The  mean  of  the  four  values  of  <p  is  38°  56'  26",  which  is  26"  less  than  the  known  true 
latitude.  This  fact  taken  in  connection  with  the  close  agreement  of  the  individual  values 
suggests  the  existence  of  an  eccentricity  correction  of  about  —  50"  for  the  part  of  the  scale 
used  in  the  observations. 

4.       ZENITH    DISTANCE    AT    ANY    HOUR    ANGLE 

63.     Theory. — It  is  desirable  to  be  able  to  determine  the  latitude  from  a 
zenith  distance  measured  when  the  object  is  so  far  from  the  meridian  that  the 


LATITUDE  FROM  ZENITH  DISTANCE  103 

formulae  for  circummeridian  altitudes  no  longer  give  convergent  results.     This 
is  readily  accomplished  by  using  the  fundamental  equation  (31)  in  the  form 

cos  £  =  «  cos  (^ — N).  (:57) 

Equation  (157)  is  the   last  of   equations  (34),  the  auxiliaries  n  and  N  being 
defined  by  the  first  and  second  of  this  group. 

64.     Procedure. — Having   determined    the   true   zenith  distance  of   the 
object,  calculate  the  hour  angle  by 

t  =  d  —  a,  (158) 

in  which  6  is  the  true  sidereal  time  of  observation.  Then  determine  n  and  N  by 

nsin  N=  sin#, 


and  <p  —  N  from 


n  cos  N  =  cos  d  cos  /, 


,T,          COS  2  ,    -    , 

—  N)  — .  (160) 


A  reference  to  the  fourth  of  (34)  shows  that  sin  (^  —  N)  must  have  the 
same  algebraic  sign  as  cos  A.  This  together  with  the  sign  of  cos  (<p  —  N}  from 
(159)  determines  the  quadrant  of  <p  —  N.  The  latitude  is  then  given  by 

(p  =  (<p  —  N)-{-N.  (161) 

Equations  (i58)-(i6i)  are  rigorous  and  apply  to  all  values  of  the  hour 
angle,  but  care  should  be  taken  to  observe  as  near  the  meridian  as  possible  in 
order  that  errors  in  z  and  6  may  not  appear  multiplied  in  the  result.  (See 
Section  56.)  A  sufficient  number  of  decimal  places  must  be  employed  to  offset 
the  fact  that  the  angle  <p  —  N  is  determined  from  its  cosine. 


Example  38.     On  1908,  Oct.  2,  at  watch 

time  8h  35™  ii"  P.M.   the  altitude 

of  Polaris  was 

found   to  be  39°  ag'.S.     (See  Ex.  31,  p.  79.) 

The  error  of 

the  watch  on  C.S.T 

.  was  +  im  45s. 

Find  the  latitude  by  equations  (158)  -(161). 

C.S.T.          8h  36™  56s 

cos  3 

8.3150 

Columbia^    21     13     14 

cos  t 

9  .  6498 

a               i     27     10 

n  cos  N 

7.9648 

t              19     46      4 

n  sin  N 

9-9999 

t                296°   31  '.o 

tan  A7" 

2-0351 

z'                 50     30.2 

N 

89°   28'.  3 

r                           i.i 

sin  N 

o.oooo 

*                 50     3i-3 

log* 

9-9999 

(J                 88    49  .  o 

cosz 

9-8033 

cos  (y>  —  N] 

I     9.8034 

The  calculated  <p  is  larger 

y  —  N 

—50°   30'.  8 

than  the  true  value  byo'.6 

<f> 

38     57.5     Ant. 

104  PRACTICAL  ASTRONOMY 

The  C.S.T.  is  converted  into  the  corresponding  Columbia  0  by  (41)  and  (58).  a  and  8 
are  from  p.  321  from  the  Ephemeris.  The  value  of  t  shows  that  Polaris  was  east  of  the  mer- 
idian at  the  time  of  the  observation,  whence  cos  A  and  sin  (y  —  N)  are  negative.  Since 
cos  (a>  —  N}  is  positive,  <p  —  N  is  in  the  fourth  quadrant. 

5.      ALTITUDE    OF    POLARIS 

65.  Theory.  —  The  peculiar  location  of  Polaris  with  respect  to  the  pole 
makes  it  possible  to  simplify  the  fundamental  latitude  equation  for  use  in 
connection  with  this  object.  Since  the  latitude  is  by  definition  equal  to  the 
altitude  of  the  north  celestial  pole,  the  problem  may  be  solved  by  finding  an 
expression  for  the  difference  in  altitude  of  the  pole  and  Polaris.  The  polar 
distance  of  Polaris  is  about  i°  n',  consequently,  the  required  difference  will 
always  be  a  small  angle.  To  this  fact  is  due  the  possibility  of  a  simplification 
of  equation  (31).  (See  Num.  Comp.  pp.  14  and  16.) 

Replacing  z  and  d  in  (31)  by  the  altitude,  /z,  and  the  north  polar  distance, 
TT,  respectively,  we  find 

sin  h  =  cos  x  sin  <f>  +  sin  TT  cos  <p  cos  /.  (162) 

If  //be  the  difference  in  altitude  of  Polaris  and  the  pole,  we  shall  have 

<?  =  h  +  H.  (163) 

Writing  h  —  tp  —  //in  (162),  and  expanding  and  solving  for  sin  H 

sin  H—  —  sin  TT  cos  t  +  tan  <p  (cos  H  —  cos  it).  (164) 

Since  at  a  maximum    . 


we  may  replace  sin  //and  sin?r  in  (164)  by  //and  ;r,  respectively,  with  an  error 
not  exceeding  0^3.     At  the  same  time  we  may  write 


COS//=1  -  >2//2,  COS7T=I  -  ^7T2, 

with  errors  which  are  still  smaller,  thus  obtaining 

H  —  —  7rcos/-i-^tan^(7r2  —  //').  (165) 

Neglecting  terms  involving  ;r3, 

H*  —  7TZ  COS2  /, 

and  substituting  H'  into  (165)  we  have 

H  —  —  TT  cos  /  +  J^TT2  tan  y?  sin2/.  (166) 

Finally,  by  (163) 


LATITUDE  FROM  POLARIS 


105 


in  which 


<p  =  h  —  TT  cos  / 


(167) 
(168) 


The  error  in  the  latitude  calculated  from  (167)  due  to  the  approximate 
form  of  the  equation  will  usually  be  less  than  2" 

The  calculation  of  K  requires  a  knowledge  of  <p  —  the  quantity  to  be 
determined;  but,  since  the  coefficient  y27t"  is  only  about  0.02,  a  rough  approx- 
imation for  the  latitude  will  answer.  The  values  of  K  may  be  derived  from 
Table  VIII  with  an  approximate  latitude  and  the  hour  angle  as  arguments. 
The  table  is  based  on  the  value  TT  =  i°  1 1' o". 

TABLE  VIII    K=%i?\xn 


t 

p-=300 

?  =  35° 

<p  =  40° 

?  =  45° 

9  =  50° 

t 

oh 

o'.oo 

o'.oo 

o'.oo 

o'.oo 

o'.oo 

I2h 

I 

0.03 

0.03 

0.04 

0.05 

0.06 

II 

2 

0.  II 

0.13 

0.15 

0.18 

O.22 

IO 

3 

0.21 

0.26 

0.31 

0-37 

0.44 

9 

4 

0.32 

0.38 

0.46 

0-55 

0.66 

8 

5 

0.40 

0.48 

0.57 

0.68 

0.82 

7 

6 

0.42 

0.51 

0.62 

0-73 

o  87 

6 

For  values  of  /  greater  than  i2h  enter  the  table  with  24h  —  t  as  argument. 

In  rough  work,  the  values  of  H  may  be  taken  directly  from  Table  IV  at 
the  end  of  the  Ephemeris  with  /,  or  24h  —  /,  as  argument.  This  table  is  calcu- 
lated with  a  mean  value  of  <p  equal  to  45°.  The  interpolated  H  added  to  the 
true  altitude  of  Polaris  will  give  the  latitude  of  points  within  10°  or  15°  of  the 
mean  latitude  of  the  table  with  an  error  not  exceeding  a  few  tenths  of  a 
minute  of  arc. 

66.  Procedure. — Having  determined  the  true  altitude,  h,  of  Polaris, 
calculate 


-  =  90°  —  d, 

(f=/i 7T  COS  t  -j-  K. 


(I69) 


in  which  6  is  the  sidereal  time  of  observation,  and  a  and  d  the  apparent  right 
ascension  and  declination,  taken  from  the  Ephemeris,  pp.  312-323.  Inter- 
polate AT"  from  Table  VIII  with  /,  or  24h  —  /,  and  an  approximate  value  of  <p  as 
arguments,  it  is  conveniently  expressed  in  minutes  of  arc. 


106  PRACTICAL  ASTRONOMY 

Example  39.     Find  the  latitude  by  equations  ("169)  from  the  data  of  Ex.  33,  p.  81. 

TV  9h  42m  10"  S  88° 49'. i  h'    39°   54:9 

J7\v  —32  TT  70-9  r  i- * 

C.  S.  T.        9  41     38  l°gr  1.8506  TTCOS/  57.0 

Columbia^?  23       i     29  cos/  9-9056  A'  0.2 

a  i  27     13  log" TT  cos/         1-7562  tp     38  57.0    ^4»5. 

1  2I  34     l6  The  calculated  u>  is  larger 


^  323  34-0 


than  the  true  value  byo'.  I. 


The  application  of  equations  (169)  to  the  data  of  Ex.  38  gives  &  =  38°  57'.5,  which  agrees 
exactly  with  the  result  obtained  by  the  formulae  of  Section  64. 

Example  4O.  Find  the  latitude  by  means  of  Table  IV  of  the  Ephemeris  from  the  data 
of  Exs.  38  and  39. 

The  hour  angle  is  to  be  calculated  as  before.  Its  value  in  both  cases  is  greater  than  i2h. 
Consequently,  H  is  to  be  interpolated  from  Table  IV,  Ephemeris^.  595,  with  2<jh  —  t  as 
argument.  We  then  find 

Ex.  38  Ex.  39 

24h  —  t  4h  I3T9  2h  25^7 

A  39°  28'.  7  39°53'.S 

H  -31-2  —57-0 

<p  38  57-5  38  56-8 

67.  Influence  of  an  error  in  time.  —  We  may  now  examine  more  closely 
the  influence  of  an  error  in  the  time  upon  the  calculated  latitude.  The  change 
is  <f>  produced  by  a  small  change  in  6  is  by  (136) 

dy  =  —  tan  A  cos  <pdd.  07°) 

For  all  of  the  preceding  methods  but  No.  4,  A  will  be  small,  a  few  degrees 
at.  most,  and  we  may  write  sin  A  in  place  of  tan  A  in  (170)  with  sufficient 
accuracy  for  the  present  purpose.  Substituting  for  sin  A  its  value  from  (33), 
and  writing  z  equal  to  the  meridian  zenith  distance,  £0,  (170)  becomes 

d<p  =  ±  cosy?  cos  £  cosec  #0  sin  tdd.  071) 

in  which  the  upper  sign  refers  to  southern  stars.    For  circummeridian  altitudes 
(171)  reduces  by  (151)  to 


O?2) 
For  Polaris  we  have  with  sufficient  approximation 

£0  =  90°  —  ^,  cos  d  =  7i  =  0.02, 

whence 

073) 


Equations  (172)  and  (173)  may  be  obtained  directly  from  (153)  and  (157) 
by  differentiating  with  respect  to  t  and  introducing  dt  =  dd,  the  small  terms 
Bn  and  K  being  disregarded. 


INFLUENCE  OF  ERROR  IN  TIME  107 

Equations  (170) -(173)  may  be  used  for  the  calculation  of  dtp  when  dd  is 
known,  or  for  the  determination  of  the  accuracy  with  which  the  time  must  be 
known  in  order  to  obtain  <f>  with  a  given  degree  of  precision.  If  dd  is  expressed 
in  seconds  of  time,  the  factor  15  must  be  introduced  into  the  right  members 
of  the  various  equations  in  order  that  dtp  may  be  expressed  in  seconds  of  arc. 

It  is  evident  that,  aside  from  the  dependence  of  d<p  upon  /,  it  also  depends 
upon  the  zenith  distance  and  declination  of  the  star,  and  that  an  error  in  the 
time  has  the  least  influence  upon  the  calculated  latitude  for  stars  near  the 
pole.  For  Polaris  the  effect  of  dd  is  always  small,  and  if  t  be  near  oh  or  I2h,  it 
will  be  very  slight  indeed,  even  though  dd  be  large. 

This  fact  taken  in  connection  with  the  simplicity  of  the  reductions  renders 
the  last  of  the  above  methods  the  most  useful  of  all  the  various  processes  that 
may  be  employed  for  the  determination  of  latitude.  The  greatest  precision, 
however,  is  attained  o^ly  by  the  method  of  a  Talcott  when  used  in  connection 
with  the  zenith  telescope. 

Example  41.  What  is  the  error  in  the  latitude  calculated  from  the  first  of  the  circum- 
meridian  altitudes  of  Ex.  37,  p  102,  on  the  assumption  that  the  watch  correction  used  was 
incorrect  by  2OS? 

By  O72)  we  nnd,  taking  the  values  of  A  and  t  from  Ex.  37 

</0  =  2os  =  30o"  *  =  7«»39''  =  i054'45" 

log  ^4         0.1744     logd<f>  1.1849 
sin  t  8.5334          <fy>  =  i5" 

log  300       2.4771 

Example  43.  How  accurately  must  the  time  be  known  in  order  that  the  altitude  of 
Polaris  given  in  Ex.  33,  p.  Si,  may  yield  a  value  of  the  latitude  uncertain  by  not  more 
than  o'.  i  ? 

By  (173)  and  the  data  in  Ex.  39,  p.  105,  we  find 

t        323°  34'- o  dy  o'.i 

sin/  °-595  0.02  sin/      0.012 

de  =  8'.  33=  33s    Atts. 


CHAPTER  VI 

THE  DETERMINATION  OF  AZIMUTH 

68.  Methods. — The  azimuth  of  a  terrestrial  mark  may  be  found  by  ob- 
serving the  difference  in  azimuth  of. the  mark  and  a  celestial  object  and  applying 
to  this  difference  the  calculated  azimuth  of  the  object  corresponding  to  the  in- 
stant of  observation.  The  methods  to  be  employed  for  the  observational  part 
of  the  process  have  been  discussed  in  detail  in  Chapter  IV.  We  have  now  to 
examine  the  means  by  which  the  azimuth  of  the  celestial  body  may  be  com- 
puted. 

A  rigorous  and  general  method  of  procedure  leading  to  the  fundamental 
equation 

cosd  sin  / 

tan  A= = — 5—  »   .  (174) 

sin  o  cos  (p  —  cos  o  sin  (p  cos  / 

was  outlined  on  page  34.  Before  proceeding  to  the  adaptation  of  this  equation 
to  the  purposes  of  calculation  it  is  desirable  to  investigate  the  conditions  under 
which  it  may  most  advantageously  be  employed.  The  calculated  azimuth  will 
depend  upon  the  right  ascension  and  declination  of  the  star,  the  time,  and  the 
latitude  of  the  place  of  observation.  The  first  two  quantities  may  be  assumed 
to  be  known  with  precision,  but  the  last  are  likely  to  be  affected  by  relatively  large 
uncertainties.  To  determine  the  influence  of  these  upon  the  calculated  azimuth, 
and  thus  derive  a  precept  for  the  choice  of  objects  to  be  observed,  we  differen- 
tiate (33),  A,  z,  and  t  being  considered  variable,  and  substitute  for  dz  its  value 
from  page  95.  Writing  at  the  same  time  dt  =.  dd  we  find  after  simplification. 

dA  =  —  cot  z  sin  A  dtp  -f-  (sin  z  sin  (p  +  cos  z  cos  ip  cos  A)  cosec  z  dd. 

If  in  Fig.  6  we  denote  the  angle  at  O  by  q,  the  expression  in  parenthesis  re- 
duces by  the  second  of  the  fundamental  formulae  of  spherical  trigonometry  to 
cos  d  cos  ^,  whence 

dA  = —  cot  2  s\nAd<p  +  cos  o  cos  q  cosec  zdd.          (175) 

In  order  that  dA  may  be  small  it  is  necessary  that  the  object  should  not 
be  near  the  zenith.  Otherwise,  the  factors  cot£  and  cosec  s  will  produce  a  multi- 
plication of  both  d<p  and  dd.  Further,  it  is  desirable  that  the  azimuth  should 
be  near  o°  or  180°,  for  when  this  is  the  case  an  error  in  the  assumed  latitude 
will  produce  but  little  effect  upon  the  calculated  azimuth.  When  the  object  is 
near  the  pole,  cos  d  will  be  small  and  the  influence  of  dd  will  be  slight ;  and  if, 
at  the  same  time,  it  be  near  elongation,  cosg  will  also  be  small,  and  the  effect 
of  dd  will  still  further  be  minimized. 

A  close  circumpolar  star  at  any  hour  angle  satisfies  these  conditions  with 
sufficient  closeness  to  render  the  influence  of  any  ordinary  errors  in  tp  and  6 
quite  insensible.  Should  the  clock  correction  be  very  uncertain,  however,  it  may 

108 


AZIMUTH  OF  THE  SUN  109 

be  desirable  to  observe  for  the  determination  of  the  azimuth  difference  of  the 
mark  and  the  star  at  or  near  the  time  of  elongation  in  order  that  the  coefficient 
of  dO  in  (175),  already  small  through  the  presence  of  cos<5,  may  be  made  still 
smaller  by  the  introduction  of  a  value  of  q  near  90°. 

Far  less  satisfactory  will  be  the  result  in  the  case  of  observations  on  the 
sun,  although  this  object  may  be  used  when  the  latitude  is  known  with  some 
precision,  provided  care  be  taken  to  observe  as  far  from  the  meridian  as  pos- 
sible. With  this  precaution  the  coefficients  in  (175)  depending  on  s  and  q  will 
have  the  values  best  adapted  for  a  minimization  of  the  errors  in  <p  and  #,  especially 
that  of  the  latter,  which  in  all  cases  is  most  to  be  feared. 

Besides  the  fundamental  equation  (174)  there  is  another  which  is  some- 
times useful,  namely,  (26).  If  the  zenith  distance  of  the  celestial  body  be 
measured  simultaneously  with  the  determination  of  the  azimuth  difference,  the 
azimuth  of  the  body  may  be  calculated  by  this  equation,  whence  the  azimuth  of 
the  mark  can  be  found  as  before.  With  this  method  of  procedure  the  latitude  of 
the  place  must  be  known,  but  the  time  does  not  enter  into  the  problem  except  as 
it  may  be  required  for  the  interpolation  of  the  declination  of  the  object  for 
the  instant  of  observation. 

To  determine  the  conditions  under  which  this  method  may  be  used  with 
advantage,  differentiate  (26)  considering  <p,  #,  and  A  as  variables.  We  thus 
find  after  simplification 

cosydA  =  cos  ^cosec  tdz  —  cot  t  dtp.  (176) 

From  this  it  appears  that  errors  in  z  and  <f>  will  have  the  least  influence 
when  t  and  q  are  as  near  90°  or  270°  as  possible.  These  conditions  cannot  both 
be  fulfilled  at  the  same  time.  But  for  circumpolar  stars  observed  near  elonga- 
tion the  magnitude  of  cosg  and  cott  in  (176)  will  be  such  that  errors  in  z  and  tp 
will  have  only  an  insignificant  influence  on  the  calculated  azimuth. 

The  consideration  of  the  preceding  results  indicates  that  we  shall  need 
adaptations  of  the  fundamental  azimuth  equations  designed  for  the  calculation  of 

1.  The  azimuth  of  the  sun. 

2.  The  azimuth  of  a  circumpolar  star  at  any  hour  angle. 

3.  Azimuth  from  an  observed  zenith  distance. 

I.      AZIMUTH  OF  THE  SUN 

69.  Theory.  —  The  first  four  equations  of  (34)  are  the  equivalent  of  (32) 
and  (33)  from  which  the  fundamental  equation  (174)  was  derived.  By  their 
combination  we  find  the  following  group  which  for  the  purposes  of  calculation 
replaces  (174). 

tan  d 


., 
tanN= 


-  , 
cos  t 


cos  N 

ta.nA=  —  —  -.  --  =77  tan  /.  (178) 

sin  (<p  —  N) 

The  quadrants  of  JV  and  A  are  determined  by  noting  that  sin  N  and  sin  A  have  the  same 
algebraic  signs  as  sin  §  and  sin  t,  respectively. 


110  PRACTICAL  ASTRONOMY 

70.     Procedure.  —  If  a  sidereal  timepiece  is  used,  calculate  t  from 


in  which  6  is  the  true  sidereal  time  of  observation,  and  a  the  sun's  apparent 
right  ascension.  If  a  mean  solar  timepiece  is  employed,  calculate  the  apparent 
solar  time  for  which  the  azimuth  difference  has  been  measured.  This  is  di- 
rectly the  hour  angle  of  the  sun.  Interpolate  d,  and  a  when  required  for  the 
calculation  of  t,  for  the  instant  of  observation.  Finally  compute  A  from  (177) 
and  (178).  Azimuth  determinations  from  solar  observations  should  be  made 
only  when  the  sun  is  far  from  the  meridian. 

2.      AZIMUTH  OF  A  CIRCUMPOLAR  STAR  AT  ANY  HOUR  ANGLE 

71.  Theory.  —  Dividing  the  numerator  and  denominator  of  (174)  by 
sin  d  cos  <p  and  writing  d  =  90°  —  TT,  we  find 

tan  TT  sec  a>  sin  / 

tanA  =  -  —  ,  (179) 

i  —  tan  TT  tan  <p  cos  / 

This  equation  may  be  replaced  by  the  following  group  which  is  arranged  with 
reference  to  the  requirements  of  calculation. 

g  =  tan  it  sec  <f>, 

k  =  tan  TT  tan  <p  =^sin  ^, 

c=          i  (180) 

i  —  h  cos  /' 
tan  A  =  —  gG  sin  t 

The  quadrant  of  A  is  determined  by  the  fact  that   sin  A  must  have  the  same  algebraic 
sign  as  sin  t. 

The  factors  g  and  h  are  constant  for  any  given  night,  and  in  approxi- 
mate work  they  may  be  considered  as  such  for  a  series  of  nights.  Moreover  h 
is  small  because  of  the  factor  tann.  G  therefore  differs  but  little  from  unity, 
and  the  values  of  logG  may  be  tabulated  with  log  h  cos  t  as  argument.  Such  a 
table,  sufficient  for  all  practical  requirements  is  given  in  Kept.  Supt.  U.  S.  Coast 
and  Geodetic  Survey,  1897-8,  pp.  399-407. 

In  case  tables  for  logG  are  not  accessible  its  values  may  be  calculated  as 
follows:  G  has  the  form  i/(i  -f-  v)  or  i/(i  —  v),  in  which  v  =  hcost,  according 
as  cost  is  negative  or  positive.  The  latter  expression  may  be  written  in  the  form 

G=i/(i—v}  =  (i+v}(i  +O(i+^)  ----         (181) 

Since  v  is  small,  the  parentheses  after  the  second  or  third  in  the  last  member 
of  (181)  will  sensibly  be  equal  to  unity.  To  find  the  value  of  logG,  therefore, 
we  must  find  the  logarithms  of  one  or  more  factors  of  the  form  (i  +  &).  For 
this  purpose  we  use  the  addition  logarithmic  table.  Since  a=i,  the  formulae 
are  (Num.  Comp.  p.  10). 


AZIMUTH  OF  A   CIRCUMPOLAR  STAR 


111 


where  B  is  to  be  interpolated  from  the  table  with  A  as  argument.     Hence 

For  cos  t  negative, 

A  =  \og(h  cost),  logG  =—B. 

For  cost  positive,  (182) 

A^  =  log  (h  cos  /),    A,  =  log  (h  cos  t)*,    A3=\og(/icost)4  .  .  .  . 


Equations  (  180)  used  in  connection  with  tables  for  log  G,  or  with  formulae 
(182)  afford  a  convenient  and  precise  method  of  calculating  the  azimuth  of 
any  of  the  close  circumpolar  stars  whose  apparent  places  are  given  in  the 
Ephemeris,  pp.  312-323. 

Equations  (180)  are  rigorous,  however,  and  for  approximate  results  they 
may  be  simplified,  especially  if  the  circumpolar  observed  is  Polaris.  For  this 
object  TC  at  the  present  time  is  i°  11',  and  for  latitudes  less  than  60°,  its  azimuth 
will  always  differ  from  180°  by  less  than  2°  3'.  We  may  therefore  write 


180° — A  =  7t 


(183) 


with  an  error  not  exceeding  2".    For  latitudes  of  45°  or  less  the  error  will  always 
be  less  than   i". 

TABLE  IX 


t 

logG 

log  G  sec  (p 

t 

0* 

o  .  0075 

o.  1167 

24h 

I 

o  0073 

0.1165 

23 

2 

0.0065 

0.1156 

22 

3 

0.0053 

0.1145 

21 

4 

0.0037 

0.1129 

20 

5 

0.0019 

O.  IIII 

19 

6 

o.oooo 

o.  1092 

18 

7 

9.9981 

o.  1072 

17 

8 

9.9963 

0.1055 

16 

9 

9.9948 

0.1039 

*S 

10 

9.9936 

o.  1028 

H 

ii 

9.9928 

O.  IO2O 

13 

12 

9.9926 

o.  1018 

12 

Further,  logG  may  be  tabulated,  as  in  Table  IX,  for  a  mean  value  of  the  lati- 
tude with  t  as  argument.  Since  <p  enters  into  G  through  h,  which  contains  the 
factor  tan  TT,  and  since  G  itself  appears  multiplied  by  TT  in  (183),  any  difference 


112  PRACTICAL  ASTRONOMY 

between  the  value  of  <p  assumed  for  the  calculation  of  the  table  and  that  cor- 
responding to  the  place  of  observation  will  have  only  a  slight  influence  on  the 
azimuth  derived  from  (183).  It  is  to  be  understood,  however,  that  the  local 
value  of  sec^>  must  be  used,  and  that  the  value  of  the  coefficient  TT  appearing 
in  (183)  must  correspond  to  the  date  of  observation.  In  case  a  number  of 
azimuth  determinations  are  to  be  made  at  a  given  station,  the  corresponding  local 
value  of  log  sec  <p  may  conveniently  be  combined  with  the  mean  values  of 
log  G.  One  can  then  interpolate  log  G  sec  <f>  directly  from  the  table. 

The  values  of  log  G  in  Table  IX  are  based  upon  y>  =  40°,  and  it  =  i°  10'. 
the  latter  of  which  is  the  mean  north  polar  distance  for  1910.0.  The  maximum 
absolute  errors  in  the  azimuth  resulting  from  the  use  of  this  table  for  various 
latitudes  are 

Latitude  30°          35°          40°          45°  50° 

Error  in  A  o'.24        o'.  12         o'.oo        o'.  15         o'.38 

The  values  of  log  G  sec  <f>  in  the  third  column  of  Table  IX  refer  to  the  lati- 
tude of  the  Laws  Observatory,  which  is  38°  57'. 

72.  Procedure. — Interpolate  a  and  3  for  the  instant  of  observation  from 
the  list  of  apparent  places  of  circumpolar  stars,  Ephemeris,  pp.  312-323.  Cal- 
culate 

71  =  90°— £,  /  =  0— a, 

where  6  is  the  true  sidereal  time  of  observation  for  which  the  azimuth  difference 
of  the  star  and  the  mark  has  been  measured.  Then : 

For  a  precise  azimuth,  calculate  A  from  (180).  The  value  of  logG  may  be 
taken  from  Kept.  Supt.  U.  S.  Coast  and  Geodetic  Survey,  1897-8,  pp.  399-407, 
or  from  some  similar  table,  with  the  argument  log  h  cos  t;  or  it  may  be  calculated 
by  means  of  (182). 

For  an  approximate  azimuth  from  Polaris,  interpolate  log  G,  or  log  G  sec  ^>, 
from  table  IX  with  t  as  argument.  Then  calculate  A  from 

A  —  i8o° — TrCsec^sinA  (184) 

If  TT  be  expressed  in  minutes  of  arc,  the  last  term  of  (184)  will  also  be 
given  in  minutes  of  arc. 

Example  43.  Determine  the  azimuth  of  the  mark  from  the  data  given  in  Ex.  34,  p.  85. 
The  latitude  of  the  place  of  observation  is  38°  56'  52". 

Equations  (180)  are  used  for  the  calculation,  the  results  for  the  two  positions  of  the 
instrument  being  reduced  separately.  The  azimuth  of  the  mark  is  found  by  subtracting  the 
difference  S — M,  taken  from  p.  85,  from  the  calculated  azimuth  of  Polaris.  The  difference 
of  the  two  values  of  M  is  not  to  be  taken  as  an  indication  of  the  precision  of  the  result,  as 
these  quantities  are  affected  by  instrumental  errors  whose  influence  is  not  eliminated  until  the 
mean  is  formed. 


113 


a      i 

7T          1° 
<       38 


log// 


25m  19* 

10'  46" 
S^  52 

0.10918 
8.31362 
9.90756 
8.42280 
8.22118 


C.  R. 

C.  L. 

0 

9h  38™  32s 

9h  49m  388 

/ 

8     13     13 

8     24     19 

t 

123°    18'   15" 

126°     4'   45" 

cost 

9-73964n 

9-77005n 

h  cos  t  =  A. 

7.96082,1 

7  ^IZSn 

B  =  log  G 

9.99605 

9-99577 

sin  / 

9.92209 

9.90752 

tan  A 

8.  34094  „ 

8.32609,, 

A 

178°  44'   38" 

178°  47'    10' 

S  —  M 

173     33     30 

173     35     34 

M 

5     ii       8 

5     ii     36 

Mean 

5°  ii' 

22" 

3-      AZIMUTH    FROM    AN    OBSERVED    ZENITH    DISTANCE 

73.     Theory. — Equation  (26)  rewritten  in  the  form 


—  cos  A  = 


sin  S  —  cos  z  sin 


sn  z  costp 
A  <C  180°  when  the  object  is  west  of  the  meridian 


(185) 


expresses  the  azimuth  as  a  function  of  d,  z,  and  <p.  If  the  zenith  distance  of  an 
object  of  known  declination  be  measured  at  a  place  of  known  latitude,  the  azi- 
muth of  the  object  can  be  calculated.  We  have  seen  from  the  differential  rela- 
tion (176)  that  the  most  advantageous  use  of  (185)  requires  that  q  and  t  be  as 
near  90°  or  270°  as  possible,  a  condition  best  fulfilled  by  circumpolar  stars. 
For  these  objects  the  azimuth  will  be  near  180°.  In  such  cases  the  solution  of 
(185)  will  be  affected  by  a  large  error  of  calculation  owing  to  the  fact  that  A 
is  derived  from  its  cosine.  On  this  account  it  is  desirable  to  transform  the 
equation  so  that  the  azimuth  may  be  determined  from  its  tangent  or  cotangent. 
This  transformation  has  already  been  made  and  the  results  are  collected  under 
(37)  along  with  the  formulae  for  the  determination  of  t  and  the  parallactic  angle, 
q,  from  the  three  sides  of  the  triangle  PZO  shown  in  Fig.  6.  Selecting  those 
relating  to  the  azimuth  we  find. 


=  z,     6  =  go0—  o, 


Check: 

cot2  j£  A  = 


(s  —  a)  -f-  (s  —  b}  +  (s  —  c}  =  s, 
sin  (s  —  c)  sin  (s  — a) 
sin  ssin  (s  —  b] 


(186) 


%  A  is  to  be  taken  in  the  ist  or  2nd  quadrant  according  as  the  object  is  west  or  east  of 
the  meridian. 

74.  Procedure. — It  is  to  be  remembered  that  the  object  observed  should 
satisfy  the  conditions  ^  —  90°  or  270°  and  /  — 6h  or  i8h  as  closely  as  possible. 
Since  the  error  of  the  measured  zenith  distance  usually  will  be  larger  than  that 
affecting  the  latitude,  the  first  of  the  above  conditions  is  the  more  important 
of  the  two.  See  equation  (176). 
8 


114  PRACTICAL  ASTRONOMY 

The  zenith  distance  determined  simultaneously  with  the  measurement  of  the 
azimuth  difference,  the  declination  for  the  instant  of  observation,  and  the  latitude 
of  the  place  of  observation  constitute  the  data  necessary  for  the  calculation  of  the 
azimuth. 

For  objects  whose  azimuths  are  not  so  near  o°  or  180°  as  to  render  the  error 
of  calculation  for  (185)  large,  we  may  calculate  A  by  this  equation.  But  for 
circumpolar  stars,  which  are  best  adapted  for  use  with  the  method  in  question,  it 
will  be  desirable  to  derive  A  by  means  of  (186).  In  any  case,  however,  (185) 
and  (186)  will  serve  as  a  mutual  control  for  testing  the  accuracy  of  the  calculated 
azimuth. 

75.  Azimuth  of  a  mark.—  Having  measured  5 — M,  the  azimuth  differ- 
ence of  the  object  and  the  mark,  and  having  determined  the  azimuth  of  the 
object  by  some  one  of  the  above  methods  we  calculate  M,  the  azimuth  of  the 
mark,  by 

(i86a) 


Example  44.     Find  the  azimuth  of  the  mark  from  the  data  given  in  Ex.  33,  p.  81. 

Since  both  the  time  and  the  altitude  of  Polaris  corresponding  to  the  instant  of  measure- 
ment of  the  azimuth  difference  of  the  star  and  the  mark  are  known,  the  reduction  may  be 
made  by  the  third  as  well  as  by  the  second  method.  The  first  column  contains  the  calcula- 
tion by  (184);  the  second,  that  by  (186).  The  value  of  t  required  for  the  first  part  is  taken 
from  Ex.  39,  p.  106. 

t  2ih  34m  16" 

t  323°  34'.o 

log;:  1.8506 

Gsec(n  0.1151 

sin/  9-7737n 

log^4N  1-7394 

^N  +54'- 9 

A  180°  54:9 

6"  —  M  174  44.1 

M  6  10.8 


Ck. 

76.  Influence  of  an  error  in  the  time. — An  uncertainty  in  the  clock 
correction,  or  any  error  in  noting  the  time  of  the  measurement  of  5  —  M,  will 
introduce  an  error  into  the  final  result,  for  the  calculated  azimuth  of  the  object 
will  not  correspond  to  the  observed  azimuth  difference.  The  magnitude  of 
this  error  for  any  given  given  error  in  6  depends  upon  the  position  of  the  star. 
Its  value  may  be  calculated  from  the  differential  relation  between  A  and  6, 

dA  =  cos  d  cos  <?  cosec  za$,  (187) 

which  is  derived  from  (175). 

Similarly,  when  the  azimuth  of  the  object  is  calculated  from  measures 
of  its  zenith  distance  there  will  be  an  uncertainty  in  the  result  due  to  the  error 
affecting  z.  The  relation  in  this  case  is,  by  (176), 


A' 

39     54-9 

r 

i.i 

a  =  z 

50      6.2 

ffj  =  * 

I     10.9 

5 

51     10.  i 

5  —  a 

i       3-9 

s  —  b 

49     59-2 

s  —  c 

o      7.0 

sin  (5  —  c) 

7.30882 

sin  (5  —  «) 

8.26920 

cosec  (5  —  £) 

0.11583 

cosec  5 

o.  10847 

cot  ^4  A 

7.90116 

A 

180°  54.8 

INFLUENCE  OF  ERROR  IN  TIME  115 

dA  =  cos  q  sec  <p  cosec  t  dz.  (l%%) 

Equations  (187)  and  (188)  may  be  used  to  estimate  the  uncertainty  in  A 
corresponding  to  a  given  uncertainty  in  6  and  z,  or  they  may  be  used  to 
determine  the  accuracy  with  which  the  time  or  the  zenith  distance  must  be 
known  in  order  to  secure  a  given  degree  of  precision  in  A. 

Usually  z  and  t  may  be  estimated  with  sufficient  precision  for  the  deriva- 
tion of  dA.  The  parallactic  angle  q  may  be  calculated  from 

sec  <p  sin  g—  sin  A  sec  «5  =  sin  /cosec  z.  (l%9) 

For  circumpolar  stars  (187)  and  (188)  may  be  simplified  as  follows:  Since 
the  azimuth  of  such  an  object  is  always  a  small  angle,  the  spherical  excess  of 
the  triangle  PZO,  Fig.  6,  page  2'6,  is  small  and  we  shall  have  approximately 
?=  1  80°  —  /,  whence 

=  —  cos/.  IO 


Further,    we  have    with    sufficient    approximation  .0  =  90°  —  <f>.     Substituting 
these  results  into  (187)  and  (188)  and  writing  cosd  =  x,  we  find 


dA=  —  Trsec  ^>  c 

dA=  —  sec^pcot/dfe.  (192) 


The   first   of  these    can    also   be    derived   from  (184)    by   differentiating   and 
writing  G=  I. 

Example  45.  The  altitude  of  the  sun  and  the  difference  of  its  azimuth  and  that  of  a 
mark  were  measured  with  an  engineer's  transit  at  the  Laws  Observatory  on  1909,  April  27. 
The  results  were  7\v  =  4h  im  n?o,  P.M.,  J7\v  —  —  im  44?5  (referred  to  C.S.T.),  h'  =  33°  19'.  6, 
6"  —  j1/=8i°  2417.  Find  the  azimuth  of  the  mark,  calculating  the  azimuth  of  the  sun  both  by 
method  i  and  method  3. 

The  computation  of  the  solar  azimuth  by  (177)  and  (178)  is  in  the  first  column;  that  for 
(186),  in  the  second.  In  the  latter  instance  the  time  is  required  only  with  such  precision  as 
as  may  be  necessary  for  the  interpolation  of  declination  from  the  Ephemeris  for  the  instant 
of  observation. 

C.S.T.  3h  59™  2655                            h>  33°  19-6 

Col.  M.S.  T.  3     50      8.2                          r—  p  1.3 

E                           2     24.7                           a  =  z  56  41.7 

;  =  Col.  A.S.T.  3     52     32.9                          b  —  -n:  76  8.9 

t                      58°     8'.  2  c  =  go  —  (p  51  3.1 

d  +13     51-*  s  9i     56-9 

tan  (5  9.39196  sin  (s  —  c)  9.81604 

cos  t  9.72255  sin  (s  —  a)  9.76132 

tan  TV  9.66941  cosec  s  0.00025 

N  25°     2'.  2  cosec  (s  —  £)  0.56498 

(p  38     56.9  co\.%A  0.07130 

tp  —  N  13     54.7  A  80°   38:2  Ck. 

cosyV  9-957I5  -S  —  M  81     24.7 

tan  t  0.20651  M  359     13.5 

cosec  ((p  —  TV)  0.61902 

tan  A  0.78268 

A  80°   38'.! 


CHAPTER  VII 

THE  DETERMINATION  OF  TIME 

77.  Methods. — The  determination  of  time  means,  practically,  finding 
the  error  of  a  timepiece.  .To  accomplish  this  the  true  time  6  or  T  is  calculated 
from  observations  on  a  star  or  the  sun  and  compared  with  the  clock  time  at 
which  the  observations  were  made.  The  required  error  is  given  by 

M=e-e\  (193) 

or 

jr=r-r,  (i94) 

according  as  the  timepiece  is  sidereal  or  mean  solar,  6'  and  T'  being  the  clock 
values  of  the  time  of  observation. 

The  fundamental  equation  for  the  determination  of  time  is 

6  =  a  +  t.  (195) 

Applied  to  any  celestial  object  this  equation  gives  the  sidereal  time,  from 
which  the  mean  solar  or  apparent  solar  time  may  be  derived  by  the  transform- 
ation processes  of  Chapter  III.  For  the  sun,  however,  the  hour  angle  /  is 
directly  the  apparent  solar  time,  and,  in  case  of  observations  on  this  object,  the 
mean  solar  time  may  be  found  from  (42)  written  in  the  form 

T=t  +  E.  (196) 

When  the  timepiece  is  solar  the  use  of  (196)  is  simpler  than  that  of  (195). 

Since  a  and  E  may  be  regarded  as  known,  the  problem  is  reduced  to  the 
determination  of  the  hour  angle  of  the  object  for  the  instant  of  observation. 
As  indicated  on  page  34  this  may  be  accomplished  by  measuring  the  zenith 
distance  of  the  object  at  a  place  of  known  latitude  and  using  equation  (38) 

or  (39)- 

The  problem  can  also  be  solved  by  determining  the  clock  time  60'  of  the 
instant  for  which  the  hour  angle  of  the  object  is  zero.  For  this  case  the 
fundamental  equation  reduces  to 


(197) 
and 

4d  =  a  —  60'.  (198) 

In  outlining  the  methods  that  may  be  employed  for  the  determination  of 
60'  it  will  be  assumed  that  the  object  is  a  star  and  that  the  timepiece  used  is 
sidereal.  The  modifications  necessary  for  the  removal  of  these  limitations  will 
be  considered  in  connection  with  the  discussion  of  the  details  presented  in  the 

following  sections. 

116 


METHODS  117 

To  determine  00'  we  may  note  the  time  0,  when  a  star  has  a  certain  zenith 
distance,  or  altitude,  east  of  the  meridian,  and,  again,  the  time  02  when  it  has 
the  same  zenith  distance  west  of  the  meridian.  Since  the  celestial  sphere 
rotates  uniformly,  we  shall  have 

0.'=X(0>+0,).  (199) 

The  method  is  known  as  that  of  equal  altitudes. 

The  clock  time  of  meridian  transit,  00',  may  also  be  determined  by  noting 
the  instant  of  passage  of  an  object  across  the  vertical  thread  of  a  transit 
instrument  mounted  so  that  the  line  of  sight  of  the  telescope  lies  in  the  plane 
of  the  meridian.  This  is  the  meridian  method  of  time  determination. 

Finally,  d0'  may  be  found  by  observing  the  transit  of  an  object  across  the 
vertical  thread  of  an  instrument  nearly  in  the  plane  of  the  meridian.  The 
application  of  a  small  correction  to  the  observed  time  depending  upon  the 
displacement  of  the  instrument  from  the  meridian  gives  the  clock  time  for 
which  ^  =  o.  In  practice  the  deviation  of  the  instrument  is  such  that  the  line 
of  sight  lies  in  the  plane  of  the  vertical  circle  passing  through  Polaris  at  a 
definite  instant.  The  process  is  accordingly  known  as  the  Polaris  vertical 
circle  method  of  time  determination.  It  is  of  special  interest  on  account  of 
the  fact  that  it  is  readily  adapted  to  a  simultaneous  determination  of  time  and 
azimuth. 

There  are  other  methods  of  determining  the  true  time,  but  those  outlined 
afford  a  sufficient  variety  to  meet  the  conditions  arising  in  practice.  We  there- 
fore proceed  to  a  detailed  consideration  of 

1.  The  zenith  distance  method. 

2.  The  method  of  equal  zenith  distances  or  altitudes. 

3.  The  meridian  method. 

4.  The  Polaris  vertical  circle  method. 

I.      THE    ZENITH    DISTANCE    METHOD 

78.  Theory. — The  formulae  necessary  for  the  calculation  of  /  from  d,  <p, 
and  £,  were  developed  in  connection  with  the  discussion  of  coordinate  trans- 
formations and  are  given  in  (38)  and  (39). 

The  resultant  error  of  observation  will  depend  upon  the  errors  affecting 
«,  (?,  tp,  and  z.  Those  in  a  and  d  we  may  disregard  as  relatively  insignificant. 
From  {136)  we  find 

dd  =  cosec  A  sec  <p  dz  —  cot  A  sec  tpdtp.  (200) 

Assuming  that  dz  and  dtp  represent  the  errors  in  z  and  tp,  and  dd  the  resultant 
error  of  observation  in  6,  it  appears  that  for  a  given  latitude  the  time  will  be 
least  affected  by  uncertainties  in  z  and  tp  when  the  azimuth  of  the  object  is 
near  90°  or  270°.  Care  should  be  taken,  therefore,  to  select  for  observation 
only  those  objects  which  are  near  the  prime  vertical. 


118 


79.  Procedure. — Having  found  the  true  zenith  distance  corresponding 
to  the  clock  time,  calculate  t  by  (38)  or  (39).  The  latter  equation  should  not 
be  used  when  the  object  is  so  near  the  meridian  that  the  interpolation  of  t  from 
its  cosine  is  rendered  uncertain. 

Observations  on  a  star:  If  the  tjmepiece  is  sidereal,  calculate  6  by  (195)^ 
and  Ad  by  (193);  if  solar,  convert  the  sidereal  time  derived  from  (195)  into  the 
corresponding  mean  solar  time  T,  and  determine  AT  from  (194),  taking  care 
that  T  is  reduced  to  the  meridian  to  which  the  clock  time  refers. 

Observations  on  the  sun:  If  the  timepiece  is  sidereal,  we  may  proceed  as 
fn  the  case  of  a  star  using  (195)  and  (193),  or  we  may  convert  the  value  of  T 
derived  from  (196)  into  the  corresponding  sidereal  time  and  then  use  (193). 
If  the  timepiece  is  solar,  calculate  T  from  (196),  reduce  its  value  to  the 
meridian  to  which  the  clock  time  refers,  and  calculate  JT^from  (194). 

Owing  to  the  change  in  the  right  ascension  and  declination  of  the  sun,  a 
knowledge  of  the  approximate  time  is  necessary  for  the  reduction  of  solar 
observations.  Should  the  error  of  the  timepiece  be  unknown,  the  interpolation 
of  a  and  <5,  or  E,  may  be  made  with  the  Greenwich  mean  time  corresponding  to 
the  clock  time  of  observation.  The  resulting  data  will  give  an  approximation 
for  the  error  of  the  clock  which,  in  general,  will  be  sufficient  for  a  precise 
interpolation  of  the  coordinates  of  the  sun.  A  repetition  of  the  calculation 
then  gives  the  final  value  of  the  clock  correction. 

Example  46.     Find  the  error  of  the  watch  from  the  measured  altitude  of  Alcyone  given 
in  Ex.  31,  p.  79. 

We  have 


21°    19'  30" 

2  30 

21       17  O 

+23     49  23 

38     56  52 


i8h  35m  i6?7 
3    42      3-2 

22       17       19.9 


C.S.T.     9    40 
Watch    9    39 


6-4 
45.0    Ans. 


The  solution  of  (38)  gives  t=  i8h35m  1658.  From  (39),  as  a  control,  we  find  i8h35m  1616, 
The  value  used  for  t  is  the  mean  of  these.  The  conversion  of  0  into  the  corresponding 
C.S.T.  is  accomplished  by  (62)  and  (41). 


2.      THE    METHOD    OF    EQUAL   ALTITUDES 

80.  Theory.  —  If  #,  and  02  be  the  sidereal  clock  times  when  a  star  has  the 
same  altitude,  or  zenith  distance,  east  and  west  of  the  meridian,  respectively, 
the  clock  time  of  meridian  transit  will  be  given  by  (199),  whence  by  (198) 


If  a  solar  timepiece  is  used  we  shall  have 


(201) 


(202) 


where  T  is  the  mean  solar  time  corresponding  to  6=  a. 

If  the  object  observed  is  the  sun,  the  above  equations  are  not  applicable 
on  account  of  the  change  in  the  declination  during  the  interval  separating  the 


TIME  FROM  EQUAL  ALTITUDES  119 

measures.  This  influence  may  be  included,  however,  by  reducing  the  observed 
times  to  what  they  would  have  been  had  the  declination  been  constant  and 
equal  to  its  value  at  the  instant  of  meridian  transit.  Since  the  change  in  d  is 
small,  the  required  corrections  may  be  found  from  the  differential  relation 
connecting  changes  in  d  with  corresponding  changes  in  /.  From  (31) 

dt  =  (tan  (f>  cosec  t  —  tan  3  cot  /)  dd,  (203) 

in  which  /is  one-half  the  interval  between  the  two  observations  expressed  in 
solar  units,  d  the  declination  for  apparent  noon,  and  dd  the  change  in  d  during 
the  interval  t.  Both  the  observed  times  will  be  too  late  by  the  quantity  dt. 
Hence,  for  solar  observations  made  with  a  sidereal  timepiece, 

AQ  =  a  —  #(0f  +  em)  +  dt.  (204) 

If  the  timepiece  is  solar,  we  have  from  (196)  and  (202),  since  /  =  0  for  the 
instant  of  meridian  transit, 

AT=E-  %  (T,  +  T.)  +  dt.  (205) 

It  is  sometimes  convenient  to  combine  afternoon  observations  with  others 
made  on  the  following  morning.  In  this  case  the  mean  of  the  observed  times 
corrected  for  the  change  in  declination  is  the  clock  time  of  lower  culmination. 
The  quantity  t  in  (203)  is  one-half  the  interval  between  the  observations 
expressed  in  solar  units  as  before;  but  d  must  be  interpolated  from  the 
Ephemeris  for  the  instant  of  the  sun's  lower  transit,  and  the  resulting  value  of 
dt  must  be  added  to  the  clock  times  of  observation.  The  expressions  for  the 
clock  correction  are 

J0  =  i2h  +  «  —  #(0,  +  6,}  —  dt,  (206) 

AT=  12"  +  E-%  (T,  +  T,) -dt,  (207) 

in  which  the  values  of  a  and  E  refer  to  the  instant  of  lower  culmination. 

81.  Procedure. — The  object  observed  should  be  near  the  prime  vertical. 
When  three  or  four  hours  east  of  the  meridian  note  the  time  of  transit  across 
the  horizontal  thread  of  the  transit  for  a  definite  reading  of  the  vertical  circle, 
most  conveniently  an  exact  degree  or  half  degree.  Change  the  reading  by 
10'  or  20'  and  note  the  time  of  transit  as  before.  Repeat  a  number  of  times, 
always  changing  the  reading  by  the  same  amount.  After  meridian  passage 
observe  the  times  of  transit  over  the  horizontal  thread  for  the  same  readings  of 
the  vertical  circle  as  before,  but  in  the  reverse  order.  If  the  sextant  is  used, 
note  the  times  of  contact  of  the  direct  and  reflected  images  for  the  same 
series  of  equidistant  readings  of  the  vernier  before  and  after  meridian  passage. 
Denote  the  means  of  the  two  series  of  times  by  0,  and  0a,  or  Tf  and  T,,  according 
as  the  timepiece  is  sidereal  or  mean  solar.  For  a  star  the  error  of  the  clock  will 
be  given  by  (201)  or  (202).  For  the  sun,  calculate  dt  by  (203),  and  the  clock  error 
by  (204)  or  (205)  in  case  the  observations  are  made  in  the  morning  and  after- 


120  PRACTICAL  ASTRONOMY 

noon  of  the  same  day,  or  by  (206)  and  (207)  when  they  are  secured  in  the 
afternoon  and  on  the  following  morning. 

Care  must  be  taken  not  to  disturb  the  instrumental  adjustments  between 
the  two  sets  of  measures.  If  these  remain  unchanged  no  correction  need  be 
applied  for  index  error,  eccentricity,  refraction,  parallax  or  semidiameter. 
This  fact  taken  in  connection  with  the  simplicity  of  the  reductions  constitutes 
the  chief  advantage  of  the  method.  It  is  subject,  however,  to  the  serious 
objection  that  an  interval  of  several  hours  must  elapse  before  the  observing 
program  can  be  completed,  with  the  danger  that  clouds  may  interfere  with  the 
second  series  of  measures. 

When  the  engineer's  transit  is  used  for  the  observations,  all  the  measures 
should  be  made  in  the  same  position  of  the  verticle  circle,  and  the  angles 
should  all  be  set  from  the  same  vernier. 

As  in  the  case  of  the  zenith  distance  method  of  time  determination,  an 
approximate  knowledge  of  the  time  is  necessary  when  the  object  observed  is 
the  sun.  If  the  clock  correction  is  quite  unknown,  this  may  be  derived  from 
the  observations  themselves  as  before.  It  is  only  necessary  to  interpolate  the 
sun's  right  ascension,  or  the  equation  of  time,  as  may  be  required,  on  the 
assumption  that  the  clock  error  is  zero.  This  approximate  result  will  lead  to 
an  approximation  for  the  error  of  the  timepiece  with  which  the  calculation 
may  be  repeated  for  the  determination  of  the  final  value. 

3.       THE    MERIDIAN    METHOD 

82.  Theory. — The  meridian  method  of  time  determination  requires 
a  transit  instrument  mounted  so  that,  when  perfectly  adjusted,  the  line  of 
sight  lies  constantly  in  the  plane  of  the  meridian,  whatever  the  elevation 
of  the  telescope.  In  order  that  this  may  be  the  case,  the  horizontal  axis 
must  coincide  with  the  intersection  of  the  planes  of  the  prime  vertical  and  the 
horizon,  and  the  line  of  sight  must  be  perpendicular  to  the  horizontal  axis. 
The  instant  of  a  star's  transit  across  the  vertical  thread  will  then  be  the  same 
as  that  of  its  meridian  passage.  Denoting  the  clock  time  of  this  instant  by  60' 
the  error  of  the  timepiece  will  be  given  by 

J0  =  «  _  00'.  (208) 

In  general,  however,  the  conditions  of  perfect  adjustment  will  not  be 
satisfied.  The  horizontal  axis  will  not  lie  exactly  in  the  plane  of  the  prime 
vertical,  nor  will  it  be  truly  horizontal.  When  produced  it  will  cut  the  celestial 
sphere  in  a  point  A,  Fig.  8,  page  65,  whose  azimuth  referred  to  the  east  point 
and  whose  altitude  we  may  denote  by  a  and  <£,  respectively.  Further,  the  line 
of  sight  will  not  be  exactly  perpendicular  to  the  horizontal  axis,  but  will  form 
with  it  an  angle  90°  -\-c.  The  quantities  a,  £,  and  c  are  known  as  the  azimuth, 
level,  and  collimation  constants,  respectively.  In  general,  therefore,  the  star 
will  not  be  on  the  meridian  at  the  instant  of  its  transit  across  the  vertical 
thread,  but  will  have  a  small  hour  angle  t  whose  value  will  depend  upon  the 
magnitude  of  the  instrumental  constants  a,  b,  and  c  and  the  position  of  the 
star.  To  obtain  the  clock  time  of  meridian  transit  we  must  subtract  t  from  the 
clock  time  of  observation,  0',  whence 


THE  MERIDIAN  METHOD  121 

d0'  =  d'  —  t,  (20g) 

and  by  (208) 

Jd  =  a  —  6'  +  t.  (210) 

The  values  of  a,  b,  and  c  can  always  be  found.  Consequently  Ad  can  be 
determined  by  (210)  when  t  has  been  expressed  as  a  function  of  the  instru- 
mental constants.  For  this  purpose  we  make  use  of  equations  (82),  (89),  and 
(33).  The  last  two  terms  of  (82)  express  the  influence  of  the  level  and  colli- 
mation  constants,  b  and  c,  upon  the  reading  of  the  horizontal  circle  of  the  engi- 
neer's transit  for  C.  R.,  or,  what  amounts  to  the  same  thing,  the  amount  by 
which  the  azimuth  difference  of  the  point  A  and  the  object  O,  when  on  the 
vertical  thread,  exceeds  90°.  The  last  two  terms  of  (89)  express  the  corres- 
ponding quantity  for  C.  L.  These  results  may  be  applied  directly  to  the 
meridian  transit  to  determine  the  azimuth  of  the  star  at  the  instant  of  its 
transit  across  the  vertical  thread.  For,  denoting  this  azimuth  by  As,  and 
assuming  that  a,  the  azimuth  of  the  point  A  referred  to  the  east  point,  is 
measured  positive  toward  the  south,  we  have  at  once 


As  =  a-\-&  cot  z±c  cosecz,  (211) 

in  which  the  upper  sign  refers  to  C.  R.,  and  the  lower  to  C.  L.  In  the  present 
case,  however,  the  positions  of  the  instrument  are  less  ambiguously  expressed 
by  circle  west  (C.  W.)  and  circle  east  (C.  E.),  respectively.  We  may  now  use 
(33)  to  determine  the  hour  angle  of  the  star  when  its  azimuth  is  equal  to  As. 
Replacing  A  in  (33)  by  As  and  writing  As  and  /  instead  of  their  sines,  which 
we  may  do  since  both  are  very  small  angles,  we  find 

t  cos  d  =  As  sin  z  (2  12) 

whence  by  (211) 

/cos  d  =  a  sin  z  -f-^cos^±  c.  (2I3) 


Equations  (211)  and  (212)  become  indeterminate  for  £  =  o,  on  account  of  the 
presence  of  A,  but  the  conditions  of  the  problem  show  that  there  can  be  no 
such  discontinuity  in  the  expression  which  gives  t  as  a  function  of  #,  £,  and  c. 
Equation  (213)  is  therefore  valid  for  2  =  0,  and  becomes  inapplicable  only  for 
stars  very  near  the  pole.  Since  the  star  is  near  the  meridian  at  the  instant  of 
observation,  z  in  (213)  may  be  replaced  by  the  meridian  zenith  distance  </>  —  d. 
Writing 

A  =  sin  (<p  —  d)secd,     B  —  cos  (^  —  d)secd,     C=secd,         (214) 
and  substituting  for  /  in  (210)  we  find 

6JB±cC.  (215) 


122  PRACTICAL  ASTRONOMY 

Equations  (214)  and  (215)  give  the  value  of  Jd  when  the  time  of  transit  6' 
across  the  vertical  thread  has  been  observed,  provided  the  instrumental  con- 
stants a,  b,  and  c  are  known.  The  quantities  A,  B,  and  C  are  called  the 
transit  factors.  Their  values  depend  only  upon  the  position  of  the  star  and, 
for  any  given  latitude,  may  be  tabulated  with  d  as  argument.  They  may  also 
be  tabulated  with  the  double  argurh'ent  S  and  z.  Tables  of  the  latter  sort  are 
to  be  found  in  Kept.  Supt.  U.  S.  Coast  and  Geodetic  Survey  1897-8,  pp.  308-319. 
These  are  applicable  for  all  points  of  observation. 

There  remains  still  the  determination  of  the  constants,  a,  b,  and  c.  The 
second  of  these  can  be  made  equal  to  zero  by  a  careful  adjustment  and  level- 
ling of  the  instrument,  or  its  value  may  be  measured  in  case  a  striding  level  is 
available.  The  azimuth  and  collimation  constants  are  best  determined  from 
the  observations  themselves.  Assuming  that  b  has  been  made  equal  to  zero, 
or  that  its  value  has  been  determined,  there  remain  in  (215)  only  three 
unknowns,  J0,  a,  and  c.  The  observation  of  any  three  stars  will  afford  three 
equations  of  condition  involving  these  quantities  from  which,  theoretically, 
their  values  may  be  determined.  Practically,  however,  the  solution  is  sim- 
plified and  rendered  more  precise  by  proceeding  as  follows: 

Suppose  that  the  transits  of  a  number  of  stars  of  various  declinations  have 
been  observed,  the  instrument  having  been  used  in  both  positions.  Consider 
the  results  for  two  of  these  having  the  same  declination  as  nearly  as  possible, 
one  observed  C.  W.,  the  other,  C.  E.  Writing 

A6'  =  a  —  d'  +  bB,  (216) 

we  have  from  (215) 

J0=J0'W+*^W+<:CW 

AO  =  M*+aAt-cC* 

Since  it  is  assumed  that  the  two  declinations  are  nearly  equal,  we  may  sup- 
pose Aw  =AR,  whence  we  find 


r    ,r     > 

L'E  -p   t^w 

which  determines  the  collimation  constant.  Should  there  be  more  than  one 
pair  of  stars  of  equal  declination,  (217)  may  be  applied  to  each.  The  mean  of 
the  resulting  values  of  c  will  then  be  accepted  as  the  final  value. 

Next,  consider  two  stars  observed  in  the  same  position  of  the  instrument 
whose  declinations  differ  as  widely  as  possible.  One  of  these  should  be  a 
northern  star,  a  circumpolar  preferably,  the  other,  a  southern  star.  Writing 

Ad"  =  A6'±cC  (218) 

we  find  for  these  objects  from  (215) 


THE  MERIDIAN  METHOD  123 


Ad  =  Ad"s  +  aA& 
whence 


Inasmuch  as  there  is  danger  of  a  change  in  the  azimuth  constant  during 
the  reversal,  a  should  be  determined  by  (219)  for  both  positions  of  the 
instrument. 

The  value  of  Ad  is  then  to  be  calculated  by 

Ad  =  Ad"-{-aA.  (220) 

The  mean  of  all  such  values  is  the  final  value  of  the  clock  correction. 

The  chief  advantage  of  the  meridian  method  of  time  determination  is  to 
be  found  in  the  fact  that  the  results  do  not  depend  upon  a  reading  of  the 
circles.  Since  the  uncertainty  of  an  observed  transit  is  considerably  less  than 
that  of  an  angle  measured  with  a  graduated  circle,  the  precision  is  relatively 
high.  It  is  the  standard  method  of  determining  time  in  observatories. 
When  carried  out  with  a  large  and  stable  instrument  mounted  permanently  in 
the  plane  of  the  meridian,  with  the  inclusion  of  certain  refinements  not  con- 
sidered in  the  preceding  sections,  it  affords  results  not  surpassed  by  those  of 
any  other  method,  either  in  precision  or  in  the  amount  of  labor  involved  in  the 
reductions. 

83.  Procedure.  —  To  place  the  instrument  in  the  meridian  we  may  make 
use  of  a  distant  object  of  known  azimuth.  Set  off  the  value  of  the  azimuth  on 
the  horizontal  circle  and  bring  the  object  on  the  vertical  thread  by  rotating  on 
the  lower  motion.  Having  clamped  the  lower  motion,  rotate  on  the  upper 
motion  until  the  reading  is  zero.  The  line  of  sight  will  then  be  approximately 
in  the  plane  of  the  meridian. 

In  case  no  object  of  known  azimuth  is  available,  Polaris  may  be  used  in- 
stead. In  this  case  the  star  is  brought  on  the  vertical  thread  at  an  instant  for 
which  its  azimuth  has  previously  been  calculated  by  (184).  With  the  ex- 
ception that  the  setting  must  be  made  at  a  definite  instant,  the  procedure  is 
the  same  as  that  for  a  distant  terrestrial  object.  The  determination  of  the 
azimuth  of  Polaris  requires  a  knowledge  of  the  approximate  time,  but  (191) 
shows  that  if  6  be  known  within  two  or  three  minutes,  the  azimuth  will  not  be 
in  error  by  more  than  one  or  two  minutes  of  arc,  which  is  sufficiently  accurate. 
In  case  the  clock  correction  is  entirely  unknown,  an  approximation  may  be 
derived  as  follows:  Set  on  Polaris  and  clamp  in  azimuth.  Then  rotate  the 
telescope  on  the  horizontal  axis  and  observe  the  transit  across  the  vertical 
thread  of  a  southern  star  of  small  zenith  distance.  Denoting  the  sidereal  clock 
time  of  transit  by  6',  the  approximate  error  of  the  timepiece  will  be  given  by 

A6  =  a  —  6'.  (221) 


124  PRACTICAL  ASTRONOMY 

Since  the  azimuth  of  Polaris  differs  but  little  from  180°,  the  line  of  sight  will 
not  deviate  greatly  from  the  plane  of  the  meridian,  especially  when  directed 
toward  points  near  the  zenith.  If  the  zenith  distance  of  the  time  star  is  not 
more  than  25°  or  30°  the  error  in  6  will  not,  ordinarily,  exceed  two  or  three 
minutes,  and  this,  as  stated  above,  is  sufficient  for  the  calculation  of  the  azimuth 
of  Polaris  with  the  precision  necessa'ry  for  the  orientation  of  the  instrument. 

The  program  will  include  the  observation  of  four  or  five  stars  in  each  posi- 
tion of  the  instrument,  reversal  being  made  at  the  middle  of  the  series.  Each 
group  should  contain  one  northern  star  to  be  used  for  the  determination  of  the 
azimuth  constant.  The  remaining  objects  should  be  southern  stars  culminating 
preferably  between  the  zenith  and  the  equator.  In  order  that  there  may  be 
sufficient  data  for  the  determination  of  the  collimation  constant,  care  should 
be  taken  to  observe  at  least  one  pair  of  stars,  one  C.  W.,  the  other,  C.  E.,  whose 
declinations  are  equal  or  nearly  so. 

For  an  instrument  whose  vertical  circle  reads  altitudes,  the  settings  which 
will  give  the  telescope  the  proper  elevation  to  bring  the  stars  into  the  field  at 
the  time  of  culmination  are  to  be  calculated  by. 

Setting  =  90°  ±(p  — <5),  (222) 

in  which  the  upper  sign  refers  to  northern  stars. 

The  star  list  with  the  setting  for  each  object  should  be  prepared  in  ad- 
vance. This  having  been  done,  the  instrument  is  to  be  levelled  and  adjusted 
in  azimuth.  Three  or  four  minutes  before  the  transit  of  the  first  star,  which 
will  occur  at  the  clock  time  a  —  J0,  set  the  vertical  circle  at  the  proper  read- 
ing, and  as  the  star  comes  into  the  field  adjust  in  altitude  until  it  moves  along 
the  horizontal  thread.  Note  the  time  of  its  transit  across  the  vertical  thread 
to  the  nearest  tenth  of  a  second.  After  one-half  the  stars  have  been  observed 
in  this  manner,  reverse  the  instrument  about  the  vertical  axis  through  180° 
and  proceed  with  the  observation  of  the  remaining  stars. 

Observations  with  the  striding  level  for  the  determination  of  b  should  be 
made  at  frequent  intervals  throughout  the  observing  program.  Level  read- 
ings increasing  toward  the  east  should  be  recorded  as  positive;  toward  the 
west,  as  negative.  If  a  striding  level  is  not  available,  the  plate  levels,  especially 
the  transverse  level,  should  be  very  carefully  adjusted  before  beginning  the 
observations  and  the  bubbles  should  be  kept  centered  during  the  measures. 

The  reduction  is  begun  by  collecting  the  right  ascension,  the  declination, 
and  the  transit  factors  for  each  star.  The  coordinates  are  to  be  interpolated 
for  the  instant  of  observation  from  the  list  of  apparent  places  in  the  Ephemeris. 
The  transit  factors  may  be  computed  by  (214),  or  better  still,  they  may  be  in- 
terpolated from  the  transit  factor  tables.  (See  page  122.)  If  the  inclination  of 
the  horizontal  axis  has  been  measured,  the  values  of  b  are  to  be  computed  by 
(113).  The  value  of  Ad'  is  then  to  be  calculated  for  each  star  by  (216).  Then 
select  two  stars  of  equal  or  nearly  equal  declination,  one  observed  C.  W.,  the 
other  C.  E.,  and  calculate  c  by  (217).  Compute  as  many  such  values  of  c  as 
there  are  pairs  of  stars  of  equal  declination,  and  form  the  mean  of  all.  With  the 
mean  value  of  c  calculate  Jd"  for  each  star  by  (218).  Then  determine  a  for 


THE  MERIDIAN  METHOD  125 

each  position  of  the  instrument  by  (219),  using  for  this  purpose  the  stars  of 
extreme  northern  and  southern  declination.  Finally  calculate  Aft  for  each 
object  by  (220).  The  mean  of  all  such  values  of  Jd  is  the  final  value  of  the 
clock  correction  corresponding  to  the  mean  of  the  observed  clock  times  of 
transit. 

In  case  the  rate  of  the  timepiece  is  large,  each  observed  6'  should  be  cor- 
rected for  rate  before  forming  the  values  of  M\  the  corrections  being  applied  in 
such  a  way  that  each  6'  becomes  what  it  would  have  been  had  all  the  observa- 
tions been  made  at  the  same  instant.  The  epoch  to  which  the  values  of  6'  are 
reduced  is  usually  the  exact  hour  or  half-hour  nearest  the  middle  of  the  series. 

Example  47.  On  1909,  May  19,  Wed.  P.  M.,  the  error  of  the  Fauth  sidereal  clock  of  the 
Laws  Observatory  was  determined  by  the  meridian  method,  the  instrument  used  being  a  Buff 
&  Buff  engineer's  transit. 

The  error  of  the  clock  was  known  to  be  approximately  -f-  7mo8.  The  azimuth  of  Polaris 
ca'culated  by  (184)  for  the  clock  time  nh5imo8  was  179°  26'$.  Vernier  A  of  the  horizontal 
circle  was  set  at  this  value,  and  at  the  clock  time  indicated  Polaris  was  brought  to  the  inter- 
section of  the  threads  by  means  of  the  lower  motion.  After  clamping,  the  upper  motion  was 
released  and  vernier  A  was  made  to  read  o°.  The  instrument  having  thus  been  placed  in  the 
meridian,  the  transits  of  four  stars  were  observed.  The  reversal  was  then  made  by  changing 
the  reading  of  vernier  A  from  o°  to  180°,  after  which  four  more  stars  were  observed.  The 
plate  levels  were  carefully  adjusted  at  the  beginning,  and  the  bubbles  were  kept  centered 
throughout  the  observations. 

The  first  of  the  tables  gives  the  observing  program  and  the  data  of  observation.  The 
various  columns  contain,  respectively,  the  number,  name,  magnitude  or  brightness,  and  the 
apparent  right  ascension  and  declination  of  the  stars;  the  setting  of  the  vertical  circle,  the 
the  observed  clock  time  of  transit,  and  the  position  of  the  circle.  The  settings  were  obtained 
by  adding  the  colatitude  51°  3'  to  the  values  of  the  declination.  For  northern  stars  this  sum 
must  be  subtracted  from  180°. 

The  second  table  contains  the  reduction  and  the  value  of  the  clock  correction  derived 
from  each  star.  The  values  of  J#'  are  obtained  by  subtracting  each  0'  from  the  correspond- 
ing a  in  accordance  with  (216).  The  third  and  fourth  columns  contain  the  values  of  the 
transit  factors  interpolated  from  the  tables  of  the  Laws  Observatory.  None  of  the  pairs 
of  stars  observed  are  suitable  for  the  determination  of  the  collimation  by  (217).  To  avoid  this 
difficulty,  approximate  values  of  the  azimuth  constant  are  derived  by  (219)  from  stars  i  and  4, 
and  6  and  8,  J#"  being  replaced  by  J&'  for  this  purpose.  The  results  are  ««•  =  -)- 252  and 
rte  =  +  454.  These  values  are  uncertain  owing  to  the  fact  that  the  influence  of  the  collimation 
has  been  neglected  in  deriving  them,  but  they  are  sufficiently  accurate  for  a  determination  of 
c  by  (2i6a),  provided  we  use  for  this  calculation  stars  whose  declinations  differ  as  little  as  pos- 
sible. Substituting  the  numerical  values  of  «,  A,  and  C  into  (2i6a)  for  stars  3  and  5,  and  2 
and  8  we  find 

A9  =  +  7m4?4  +  i  -04^  J#  =  +  7m4?4  +  i  -o.<5c 

j#  =  +  7  4.6— i.ooc  j0  =  -f  7   5.0— i.ooc 

These  two  sets  of  equations  give  for  c,  -\-  0510  and  -)- °-27>  respectively.  The  mean,  -f-°fi9,  is 
accepted  as  the  value  of  the  collimation  constant.  Multiplying  this  by  the  value  of  C  for 
each  star  gives  the  corrections  for  collimation  contained  in  the  fifth  column.  The  combina- 
tion of  these  with  the  value  of  J$'  gives  the  quantities  in  the  column  headed  J#".  It  should 
be  noted  that  the  algebraic  sign  of  the  collimation  correction  changes  with  the  reversal  of  the 
instrument.  The  azimuth  constant  is  now  redetermined  for  each  position  of  the  circle,  using 
for  this  purpose  the  value  of  J#"  for  stars  i  and  4,  and  6  and  8.  The  results  are  «w  =  +  2138 
and  «e=r-^4;i6.  From  these  we  find  the  values  of  the  azimuth  corrections  aA,  which, 
added  to  the  values  of  J#"  in  accordance  with  (220),  give  J#,  the  clock  correction  for  each  star. 


126 


PRACTICAL  ASTRONOMY 


The  last  column  contains  the  weight  assigned  to  each  result  in  forming  the  mean  value  of 
the  clock  correction.  The  mean  J0  for  the  southern  stars  is  the  same  for  each  position  of 
the  instrument,  which  shows  that  the  influence  of  the  collimation  has  been  satisfactorily 
eliminated.  It  should  be  noted,  as  a  control  upon  the  calculation  of  the  azimuth  constant, 
that  the  values  of  AO  for  each  pair  of  azimuth  stars  must  agree  within  one  unit  of  the  last 
place  of  decimals.  In  the  present  case  the.agreement  is  exact  for  both  pairs. 


No. 

Star 

Mag. 

a 

d 

Setting 

r 

Circle 

i 

e  Corvi 

3-2 

I2h    5x127:3 

—  22°    7' 

28°  56'  S 

nh58m25«8 

W 

2 

Y  Corvi 

2.7 

II        8.2 

—  17       2 

34     i    S 

12     4     5-7 

W 

3 

<5*  Corvi 

3-1 

25    10.  i 

—  16     i 

35     2   S 

18     7.6 

W 

4 

x  Draconis 

3-8 

29  39.0 

+  70    17 

58   40  N 

22     32.0 

W 

5 

•r-  Virginis 

2.9 

37     3-9 

—    o  57 

50     6  S 

3°     2-1 

E 

6 

32*  Camelop. 

5-2 

48  36.1 

+  83   54 

45     3  N 

4I        2 

E 

7 

e  Virginis 

3-i 

12  57   39-8 

+  n    27 

62   30   S 

50   37-i 

E 

8 

0  Virginis 

4.6 

13    5    15-2 

—    5     3 

46     o  S 

58  13-3 

E 

No. 

AO' 

A 

C 

cC 

JT 

aA 

AO 

Wt. 

i 

+  7»i!5 

+  o-94 

-|-  i.  08 

+  052 

+  7mi!7 

+     252 

+  7m3.9 

i 

2 

25 

+  0.87 

1.05 

+  0.2 

2.7 

+     2.1 

4-8 

i 

3 

2-5 

+  0.85 

i  .04 

+  O.2 

2.7 

+      2.0 

4-7 

i 

4 

7.0 

-  1-54 

2.96 

+  0.6 

7.6 

—    3-7 

3-9 

0 

5 

1.8 

-(-0.64 

I  .00 

—  O.2 

1.6 

+    2.7 

4-3 

i 

6 

34-i 

-6.65 

9.41 

—  1.8 

32-3 

—  27.7 

4-6 

o 

7 

2.7 

+  0-47 

i  .02 

—  0.2 

2-5 

+      2.0 

4-5 

i 

8 

1.9 

+  0.70 

+   I  .00 

—  O.2 

1-7 

+      2.9 

4.6 

i 

At  0'= 


4.      THE    POLARIS    VERTICAL    CIRCLE    METHOD 
SIMULTANEOUS    DETERMINATION    OF   TIME   AND    AZIMUTH 

84.  Theory. — In  the  method  now  to  be  discussed  the  transits  of  stars 
are  observed  across  the  vertical  circle  passing  through  Polaris,  the  instrument 
being  adjusted  with  reference  to  the  plane  of  this  circle  by  bringing  Polaris  on 
the  vertical  thread  immediately  before  each  transit.  Since  the  azimuth  of 
Polaris  is  always  a  small  angle,  that  of  each  time  star  at  the  instant  of  its 
observation  will  also  be  small.  The  conditions  do  not  therefore  differ  essen- 
tially from  those  in  the  meridian  method,  and  the  clock  correction  may  be 
calculated  by  (215)  as  before.  The  only  question  to  be  considered  is  whether 
the  approximations  introduced  in  deriving  this  equation  are  justifiable  in  view 
of  the  fact  that  the  value  of  a  in  the  vertical  circle  method  may  amount  to  i° 
or  2°,  while  with  the  meridian  method  it  need  not  exceed  i'  or  2'.  It  can  be 
shown  that,  when  it  is  a  question  of  hundredths  of  a  second  of  time  in  the 


THE   VERTICAL  CIRCLE  METHOD  127 

final  result,  (215)  is  insufficient;  but  for  those  cases  in  which  an  uncertainty  of 
one  or  two  tenths  of  a  second  is  permissible,  the  approximation  is  ample. 

In  the  meridian  method  both  a  and  c  are  determined  from  the  observations 
themselves.  Here  we  determine  c  as  before,  but  a  is  to  be  calculated  from  the 
known  position  of  Polaris*  The  azimuth  constant  will  nearly  equal  the 
azimuth  of  Polaris  measured  from  the  north  point  positive  toward  the  east  at 
the  instant  of  setting,  but  not  exactly,  owing  to  the  presence  of  the  instru- 
mental constants  b  and  c.  If  a0  represent  the  azimuth  of  Polaris  defined  as 
above,  we  have  by  (82)  and  (89) 

a  =  a0  +  b  cot  z0  ±  c  cosec  z0, 

z0  being  the  zenith  distance  of  Polaris.     Since  b  and  c  are  very  small,  z0  may  be 
replaced  by  90  —  ^,  whence 


tany?  ±  ^rsec  <p.  (223) 

Substituting  (223)  into  (215)  and  writing 

B'  =  A  tan  <p  +  B,         C  =  A  sec  <p  +  C,  (224) 

we  have 

cC,  (225) 


where,  as  before,  the  upper  sign  refers  to  C.  W.     Equation  (225)  is  the  same 
in  form  as  (215);  but  its  solution  is  slightly  different,  for  (184)  gives 


a0  =  —  TT  £sec^>  sin  4i  (226) 

which  may  be  used  for  the  calculation  of  a0.  This  leaves  in  (225)  only  two 
unknowns,  J#  and  c,  and  the  observation  of  any  two  time  stars  therefore 
affords  the  data  necessary  for  a  complete  solution  of  the  problem.  For  the 
sake  of  precision  one  of  these  should  be  observed  C.  W.,  the  other,  C.  E.  To 
determine  c  write 

Jd'  =  a  —  d'  +  a0A  +  bB'.  (227) 

We  then  find  from  (225) 

jo=je'w+cCv, 

40  =  40',—  cCn 

whence 

Jtf',-J0'» 

c  —    r'     \    r>  —  •  (22°) 

I*    E      I      W    W 

There  is  here  no  necessity  for  an  equality  in  declination  of  the  two  stars  as 
in  the  case  of  the  meridian  method,  for  the  influence  of  the  azimuth  is  in  this 


128  PRACTICAL  ASTRONOMY 

case  included  in  J0'.  Having  found  c  from  (228)  we  calculate  Ad  from  (225) 
written  in  the  form 

M=A6'±cC.  (229) 

The  factor  A  in  (227)  is  the  s^me  as  that  in  (215),  but  it  must  be  more 
accurately  known  than  in  the  meridian  method,  on  account  of  the  magnitude 
of  a0.  The  quantities  B'  and  C  are  easily  reduced  by  (214)  to 

B'  =  sec  y,        C  =  E  -\-  tan  y,  (230) 

in  which 

E  =  secd  —  tan  o.  (231) 

The  values  of  E  may  be  taken  from  Table  X  with  d  as  argument,  whence  C  may 
be  found  by  the  simple  addition  of  tan^>.  For  any  given  latitude  C'  itself 
may  be  tabulated  with  d  as  argument.  The  third  column  of  Table  X  contains 
such  a  series  of  values  for  the  latitude  of  the  Laws  Observatory,  viz.,  38°  57'. 

The  vertical  circle  method  is  easily  adapted  to  a  simultaneous  determi- 
nation of  time  and  azimuth.  If  the  horizontal  circle  be  read  at  the  instant  of 
setting  on  Polaris,  and  if  in  addition,  readings  be  taken  on  a  mark,  the  azimuth 
of  the  mark  will  be  given  at  once;  for  the  azimuth  of  Polaris  is  calculated  in 
the  course  of  the  reduction  of  the  observations  for  time,  and  the  horizontal 
circle  readings  give  the  azimuth  difference  of  the  star  and  the  mark.  Since  a0 
is  measured  from  the  north  point  positive  toward  the  east,  the  azimuth  of  the 
mark  measured  in  the  conventional  manner  will  be 

Am  =  M—S-\-a0—  1 80°  (232) 

in  which  5  and  M  are  the  means  of  the  horizontal  circle  readings  on  the  star 
•and  the  mark,  respectively;  and  a0,  the  mean  of  the  calculated  azimuths  of 
Polaris. 

The  vertical  circle  method  of  time  determination,  like  that  of  the  meridian 
method,  is  not  dependent  upon  the  reading  of  graduated  circles,  and  in  conse- 
quence, yields  results  of  a  relatively  high  degree  of  precision.  It  possesses 
the  further  advantage  that  no  preliminary  adjustment  in  the  plane  of  the 
meridian  is  necessary.  It  is  especially  valuable  for  use  with  unstable  instru- 
ments, for  the  constancy  of  the  quantities  a,  b,  and  c  is  assumed  for  only  a 
very  short  interval,  much  less  than  in  the  meridian  method.  It  is  necessary 
that  the  azimuth  and  level  constants  remain  unchanged  only  during  the 
interval  separating  the  setting  on  Polaris  and  the  transit  of  the  time  star 
immediately  following,  and  this  need  not  exceed  two  or  three  minutes.  More- 
over, each  set  of  two  time  stars  is  complete  in  itself  and  gives  a  complete 
determination  of  the  error  of  the  timepiece. 

The  instrument  used  should  be  carefully  constructed,  however,  for  any 
irregularity  in  the  form  of  the  pivots  is  likely  to  produce  serious  errors  in  the 
results. 


THE  VERTICAL  CIRCLE  METHOD  129 

85.  Procedure.  —  The  instrument  is  carefully  levelled,  and  three  or  four 
minutes  before  the  transit  of  a  southern  star  across  the  vertical  circle  through 
Polaris,  the  telescope  is  turned  to  the  north,  and  Polaris  itself  is  brought  to 
the  intersection  of  the  vertical  and  horizontal  threads.  The  instrument  is 
clamped  in  azimuth  and  the  sidereal  time  of  setting,  00,  is  noted.  The  tele- 
scope is  then  rotated  about  the  horizontal  axis  until  its  position  is  such  that 
the  southern  or  time  star  will  pass  through  the  field  of  view.  The  transit 
of  the  time  star  is  observed,  and  the  entire  process  is  then  repeated  for  a 
second  time  star,  with  the  instrument  in  the  reversed  position.  The  data  thus 
obtained  constitute  a  set  and  permit  a  determination  of  the  error  of  the  time- 
piece. 

If  a  simultaneous  determination  of  time  and  azimuth  is  required,  the 
program  for  a  set  will  be 

Set  on  the  mark  and  read  the  H.  C. 

Set  on  Polaris,  note  the  time,  and  read  the  H.  C.  \  C.  W. 

Observe  the  transit  of  the  time  star. 

Set  on  Polaris,  note  the  time,  and  read  the  H.  C.  1 

Observe  the  transit  of  the  time  star.  L  C.  E. 

Set  on  the  mark  and  read  the  H.  C. 

in  which  C.  VV.  and  C.  E.  are  to  be  interpreted  as  meaning  that  if  the  instru- 
ment be  turned  from  the  mark  to  the  north  by  rotating  about  the  vertical 
axis,  the  vertical  circle  will  then  be  west  or  east,  respectively.  The  plate  levels 
should  be  carefully  watched,  and  if  there  is  any  evidence  of  creeping,  the  in- 
strument should  be  relevelled. 

The  observing  list  with  the  settings  for  the  time  stars  should  be  prepared 
in  advance.  It  is  also  desirable,  in  order  to  save  time  in  observing  and  to  avoid 
errors  in  the  identification  of  the  stars,  to  calculate  in  advance  the  approxim- 
ate times  of  transit.  Disregarding  the  errors  in  level  and  collimation  we  have 
from  (225) 

0'  =  «-f<v4  —  J0  (233) 

in  which  J6  represents  an  approximate  value  of  the  clock  correction.  To  de- 
rive a  value  for  the  term  a0A  we  combine  equation  (226)  with  the  value  of  A 
from  (214),  and  write 


We  thus  find 

a0A  =  P(tano  —  tan  ^),  (234) 

in  which 

P  —  4^7  sin  (00  —  ih3Om).  (235) 

Since  a0A  need  be  known  only  very  roughly,  we  may  use  a  constant  value 
for  00,  choosing  for  this  purpose    the  sidereal   time  corresponding  approxi- 
mately to  the  middle  of  the  observing  program. 
9 


130  PRACTICAL  ASTRONOMY 

P  having  been  calculated  from  (235)  we  find  the  value  of  a0A  for  each  time 
star  from  (234)  by  introducing  the  corresponding  value  of  8.  One  or  two 
places  of  decimals  are  ample  for  the  calculation. 

The  observations  having  been  secured,  the  first  step  in  the  reduction  is 
the  determination  of  an  approximation  for  the  clock  correction  of  sufficient 
accuracy  for  the  calculation  of  the  azimuth  of  Polaris.  Neglecting  errors  in 
level  and  collimation  we  have  from  (225) 

J60  =  a—0'  +  a0A,  (236) 

which  applied  to  the  time  star  transiting  nearest  the  zenith  will  give  the 
required  approximation.  For  the  term  a0A  we  may  introduce  the  value  calcu- 
lated by  (234)  in  preparing  the  observing  list.  Collecting  results  we  have  the 
following  notation  and  formulae: 

«0,  TT,  and  a,  d  are  the  coordinates  of  Polaris  and  the  time  star, 

respectively; 

60  and  d\  the  sidereal  clock  times,  respectively,  of  their  observation; 
5  and  M,  the  readings  of  the  horizontal  circle  for  settings  on  Polaris 

and  the  mark,  respectively; 
Am,  the  azimuth  of  the  mark  measured  from  the  south,  positive 

toward  the  west; 
J0,  the  error  of  the  timepiece,  and  J#0,  an  approximation  for  this 

quantity. 

t0  =  60-}-dd0  —  «ov  a0  =  —  TiG  sec  <p  sin  /„, 

A  =  sin  (<f>  —  d)secd,  C'  =  tan(/>  -\-  E, 

Ad'  =  a  —  d'Jra0A+bszc<p  (237) 

_  Jfl'—  -J0' 

c 


C  '  +  C'    ' 


Log  G  or  log  £sec  <p  is  to  be  taken  from  Table  IX,  which  is  reprinted  here 
for  convenience,  with  the  argument  /0;  E  or  C',  from  Table  X  with  the  argu- 
ment d.  The  subscripts  w.  and  ^  refer  to  observations  made  circle  west  and 
circle  east,  respectively.  Finally,  calculate 

Am  =  %  [Me  —  (S-a0)e  +  Mw  —  (S—  a0)w-]-  180°,        (238) 


where  the  subscripts  attached  to  M  refer  to  settings  made  with  the  instrument 
in  such  a  position  that  if  turned  toward  the  north  by  rotation  about  the  verti- 
cal axis,  the  circle  would  then  be  west  or  east,  respectively,  according  to  the 
subscript. 

For  the  determination  of  the  error  of  the  clock,  a0  should  be  expressed  in 
seconds  of  time;  for  the  determination  of  the  azimuth,  in  minutes  of  arc.  The 
values  of  A  are  needed  to  four  places  of  decimals,  and  when  once  obtained, 
should  be  preserved,  since,  for  a  given  latitude,  they  may  be  used  unchanged 


THE   VERTICAL  CIRCLE  METHOD 


131 


for  several  months.  If  the  collimation  is  known  to  be  small  and  the  declina- 
tions of  the  two  time  stars  do  not  differ  too  greatly,  it  will  be  sufficient  to 
lake  the  mean  of  the  values  of  J0'  for  C.  W.  and  C.  E.  as  the  error  of  the 
timepiece. 


TABLE  IX,  1910.0 


'o 

log  G 

log  G  sec  (f> 

to 

Oh 

0.0075 

o.  1167 

24h 

I 

0.0073 

0.1165 

23 

2 

0.0065 

0.1156 

22 

3 

0.0053 

0.1145 

21 

4 

0.0037 

o.  1129 

2O 

5 

0.0019 

O.  IIII 

19 

6 

o  oooo 

0.1092 

18 

7 

9.9981 

o.  1072 

17 

8 

9.9963 

0.1055 

16 

9 

9.9948 

o  1039 

15 

10 

9.9936 

o.  1028 

H 

ii 

9.9928 

O.  IO2O 

13 

12 

9.9926 

o.  1018 

12 

TABLE  X 


d 

E 

C' 

+  30° 

0.58 

i-39 

+  25 

0.64 

i-45 

-f-  20 

0.70 

'•5i 

+  15 

0.77 

1.58 

+  10 

0.84 

1.65 

+  5 

0.92 

1.72 

o 

I  .00 

i.  Si 

—  5 

1.09 

i  .90 

—  10 

1.19 

2.00 

—  15 

1.30 

2.  II 

—  20 

i-43 

2.24 

—  25 

i-57 

2.38 

—  3° 

i-73 

2-54 

Example  48.  On  1909,  May  19,  immediately  after  securing  the  meridian  observations 
given  in  Ex.  47,  a  simultaneous  determination  of  time  and  azimuth  was  made  by  the  Polaris 
vertical  circle  method,  the  instrument  used  being  the  same  as  that  employed  for  the  meridian 
observations.  The  stars  observed  were 


Object  Mag.  R.  A. 

Polaris  2.2  ih  25° 

a  Virginis  i .  i  13     20 

£  Virginis  3.6  13     30 


Dec.  Setting 

34;         +  88°  49'  3"      — 

24.9         —  IO      41  40      22 

4-4       —    o      8  50    55 


During  the  observations  the  vertical  circle  of  Polaris  was  so  nearly  in  coincidence  with 
the  meridian  that  no  special  calculation  of  the  instant  of  transit  of  the  time  stars  across  this 
circle  was  necessary.  The  record  of  the  observations  is  as  follows: 

Horizontal  Circle 


Object 
Mark 
Polaris 
a  Virginis 
Polaris 
£  Virginis 
Mark 

Fauth 

Clk. 

Ver.  A 

7°  46'.o 
179    56-5 

Ver.  B      ( 
187    46  '.o 
359    56.5 

Circle 
W 
W 
W 
E 
E 
E 

13     10 

13 
16 

22 

20 
10.5 
2 

57-2 

359 

187 

58.5 
45-5 

179 

58 

•5 

7 

45 

-5 

We  have  n  —  7o'.95,  log  n  =  1.8510.  For  the  calculation  of  the  azimuth  of  Polaris  we  use 
the  approximate  clock  correction  A00  =  +  7mo",  whence  a0 — d00=  Ihi8m34§.  The  combination 
of  this  with  Q  in  accordance  with  (237)  gives  t  . 


132 


PRACTICAL  ASTRO NOMT 


The  azimuth  a  whose  logarithm  is  given  in  the  fourth  line  is  expressed  in  minutes  of  arc. 
Since  the  correction  a  A  must  be  expressed  in  seconds  of  time  the  logarithm  of  4,  viz.,  0.6020, 
is  also  included  when  log  a  and  the  two  logarithms  immediately  following  it  are  added  to 
form  log  a  A.  The  final  value  of  the  clock  correction  is  in  satisfactory  agreement  with  that 
found  in  Ex.  47. 

a  Virg-fnis,  C.W.     £  Virginis,  C.  E. 


sin  /, 

G  &QC<p 

log  a, 
sec  d 

a0A 

a  —  0 

Ad' 

C' 

cC' 

AO 


M 


nh  51™  46' 

nh  57m  28* 

8-5553 

8  0435 

0.1018 

0.1018 

o.5o8in 

9  •  9963n 

J)           9.8819 

9.7996 

0.0076 

0.0000 

0-9996n 

°-3979n 

—  1050 

—  255 

+  7     H-4 

+  7      7.2 

+  7      4-4 

+  7      4-7 

2.0 

1.8 

+  0.2 

—  O.I 

+  7m4! 

!6 

—  3'.  2 

—  I'.Q 

179    59-7 

359     59-5 

7    46.0 

187    45-5 

+  7°  46'.  i 

PAGE 
3. 


LINE 
29, 


28,  14  of  arguments, 

3*>  H> 

37.  ii, 

37,  17, 

37,  17. 

37,  18, 

39,  last, 

40,  2,  Ex.  ii,    " 

41,  2,  Ex.  13, 

42,  4  and  5,  Sec.  26, 
60,  last, 

64,  eq.  (72), 

70,  17. 

73, 

75,  prec.  eq.  (117), 


81,         4,  Ex.  33, 

85, 

96,    prec.  eq.  (141), 

98, 


ERRATA 


Many  nebulae  show  continuous  spectra,  indicating  that  they   may  not 

be  wholly  gaseous  in  constitution. 
for  cos  z  co  <p  read  cos  z  cos  <p. 
for  or  read  and. 
for  o  to  24  read  o  to  23. 
for  o  to  24  read  o  to  23. 
for  o  to  12  read  I  to  12. 
for  o  to  12  read  I  to  12. 

the  equation  number  refers  to  both  equations. 
for  lime  read  time. 
for  apparant  read  apparent. 

interchange  7S  and  7m. 
for  O.O5d  read  o.o$d.. 
for  — Ez  read  -\-  E2. 
for  i8o°-(-c  read  180°,  approximately. 

in  form  for  record  of  level  observations:  in  last  columnj  for  r'  and  r", 

read  /'  and  /";  for,  Sd,  read  SD. 
for  (112)  read  (116). 

in  Ex.  31,  add:     Alcyone   was   east   of   the    meridian  at   the  time   of 

observation. 
for  Thursday  read  Tuesday. 

in  Ex.  34,  add:     The  observations  were  made  at  the  Laws  Observatory. 
for  obervation  read  observation 

first  eq.,_/br  z'  read  z'n. 

ecl-  (H3)j  for  r  read  rs. 


INDEX 


(THE   NUMBEBS   REFER   TO  PAGES.) 


Aberration:  defined,  14;  diurnal,  15. 

Almucanters,    9. 

Altitude:  defined,  10;  measurement  of  with 
engineer's  transit,  77-80;  with  sextant, 
91-94. 

Altitude  circles,  9. 

Apparent   place,   16. 

Apparent  solar  time:  defined,  36;  converted 
into  mean  solar  time,  40. 

Arguments:    arrangement  of,  27. 

Artificial  horizon,  59. 

Asteroids,  1. 

Azimuth:  relation  to  axis  of  celestial 
sphere,  7;  defined,  10;  calculation  of  from 
latitude,  declination  and  zenith  distance, 
31-34;  conditions  for  precise  determina- 
tion of,  108;  from  the  sun,  109-110;  from 
circumpolar  star,  110;  from  measured 
zenith  distance,  113-114;  of  mark,  114; 
influence  of  error  in  time  on,  114-115; 
simultaneously  determined  with  time, 
126-132. 

Azimuth  and  zenith  distance  transformed 
into  hour  angle  and  declination,  25-28; 
into  right  ascension  and  declination,  31. 

Calendar:    Julian   and   Gregorian,   38-39. 

Cardinal  points,  9. 

Celestial  equator,  9. 

Celestial  sphere:  defined,  4;  relation  of  its 
position  to  latitude,  azimuth  and  time, 
7. 

Chronometer:  see  timepieces. 

Circummeridian  altitudes:  latitude  from, 
99-102. 

Clock:  see  timepieces. 

Coincident  beats,  53. 

Collimation:   error,  64;   constant,  120. 

Common  year,  39. 

Coordinates:  necessity  for,  8;  primary  and 
secondary,  9;  systems  of,  10;  relative  po- 
sition of  reference  circles,  23-25;  trans- 
formations of,  25-31. 

Copernican  system,  4,  6. 

Date:  calendar  and  civil,  37. 

Day:  apparent  solar,  36;  mean  solar,  37; 
sidereal,  37. 

Declination:  circles  of,  9;  defined,  10. 

Dip  of  horizon,  91. 

Diurnal  motion,   4. 


Eccentricity:  defined,  63;  determination  of 
for  sextant,  90. 

Ecliptic,  5,  9. 

Engineer's  transit:  historical,  61;  condi- 
tions, satisfied  by,  62-63;  theory  of,  64-71; 
measurement  of  vertical  angles,  77-80; 
of  horizontal  angles,  80-85. 

Ephemeris,  12. 

Equal   altitudes:    time  from,    117,   118-120. 

Equation  of  time,  40. 

Equator:    celestial,   9;    mean,   13. 

Equinox:  vernal  and  autumnal,  9;  mean, 
13;  precession  of,  13. 

Error  of  timepiece,  51. 

Fundamental  formulae  of  spherical  trigo- 
nometry, 21-23. 

Gregorian  calendar,  39. 

Horizon:  defined,  8;  artificial,  59;  dip  of, 
91. 

Horizontal  angles:  measurement  of,  80-81; 
by  repetitions,  81-85. 

Hour  angle:  defined,  10;  transformed  into 
right  ascension,  29-30;  calculation  of 
from  latitude,  declination  and  zenith 
distance,  31-32. 

Hour  angle  and  declination  transformed 
into  azimuth  and  zenith  distance,  29. 

Hour  circles,  9. 

Index  error:  of  engineer's  transit,  &8;  of 
sextant,  89. 

Julian  calendar,   39. 

Julian  year,  39. 

Latitude:  relation  to  axis  of  celestial 
sphere,  7;  defined,  24;  fundamental  for- 
mulae for,  32-34;  conditions  for  precise 
determination  of,  95;  calculated  from 
meridian  zenith  distance,  96;  by  Tal- 
cott's  method,  97-99;  from  circummeri- 
dian  altitudes,  99-102;  from  zenith  dis- 
tance at  any  hour  angle,  102-103;  from 
altitude  of  Polaris,  104-105;  influence  of 
an  error  in  time  upon,  106. 

Leap  year,  39. 

Least  reading  of  vernier,  59-60. 

Level:  error  of,  64;  theory  of,  71-72;  pre- 
cepts for  use  of,  72;  value  of  one  dlvifc 
sion,  73-76;  constant,  120. 

Longitude,  7. 

Mean  equator,  13. 


133 


134 


INDEX 


Mean  equinox,  13. 

Mean  noon:   defined,  37;   sidereal  time  of, 

47. 

Mean  place,  16. 
Mean  solar  day,  37. 
Mean   solar   time:    defined,    36;    converted 

into  apparent  solar  time,  40;    converted 

into  sidereal  time,  47-48. 
Mean  sun:   defined,  36;  right  ascension  of, 

44-46. 

Meridian:   defined,  8;   reduction  to,  100. 
Meridian    method   of    time    determination, 

117,   120-126. 
Meridian   zenith   distance:    latitude    from, 

96. 

Method  of  repetitions,  81-85. 
Nadir,  8. 
Nebulae,  3. 

Noon:  apparent,  36;  mean,  37;  sidereal,  38. 
Nutation,    13. 
Parallactic   angle,   31. 
Parallax:    defined,   14;    theory,   18-20. 
Planets:   names,  1;   relative  distances  and 

diameters,  2. 
Polar  distance,  10. 
Polaris:    latitude    from,    104-105;    azimuth 

from,  110-114. 

Polaris  vertical  circle  method  of  time  de- 
termination, 117,  126-132. 
Poles  of  celestial  sphere,  8. 
Precession,  13,  15. 
Prime  vertical,  9. 
Proper  motion,  14,  15. 
Ptolemaic  system,  4,  6. 
Rate  of  timepiece,  51. 
Reduction  to  the  meridian,  100. 
Refraction:     defined,     12;     discussion     of, 

16-18;  table,  20;  differential,  98. 
Repetitions:   method  of,  81-85. 
Residuals,  57. 
Right  ascension:   defined,  10;   transformed 

into    hour   angle,    29-30;    of   mean    sun, 

44-46. 

Right    ascension    and    declination    trans- 
formed into  azimuth  and  zenith  distance, 

31. 

Rotation:  diurnal,  4. 
Semidiameter,  77,  93. 
Sextant:  historical,  85;  theory,  86-87; 

adjustments,  88-89;   index  correction,  89; 


eccentricity,  90;   precepts  for  use  of,  91. 

Sidereal  day,  37. 

Sidereal  noon,  38. 

Sidereal  time:  defined,  25,  37;  converted 
into  mean  solar  time,  48-49. 

Solar  system:  parts,  1;  model  of,  2. 

Solstices,  9. 

Spherical  trigonometry:  fundamental  for- 
mulae, 21-23. 

Standard  time,  37. 

Stars:    motions  of,   3,   14,   15. 

Stellar  system,  1. 

Successive  approximations,   34,  118,  120. 

Sun:  annual  motion  of,  5;  parallax  of 
19-20. 

Talcott's  method,   97-99. 

Time:  relation  to  celestial  sphere,  7;  fun- 
damental formulae  for,  32-34;  basis  of 
measurement,  36;  different  kinds,  36; 
distribution  of,  38;  difference  in  two  lo- 
cal times,  39;  apparent  solar  into  mean 
solar  and  vice  versa,  40;  relation  be- 
tween units,  42;  mean  solar  into  side- 
real, 47;  sidereal  into  mean  solar,  48; 
methods  of  determining,  116-117;  from 
zenith  distance,  117-118;  from  equal  al- 
titudes, 118-120;  meridian  method,  120- 
126;  Polaris  vertical  circle  method, 
126-132. 

Timepieces:  historical,  50-51;  error  and 
rate  of,  51;  comparison  of,  52-58;  care 
of,  58. 

Transit:    see  engineer's  transit. 

Transit  factors,  122. 

Tropical  year,  38. 

True  place,  16. 

True  solar  time:   see  apparent  solar  time. 

Vernier:  theory,  59-60;  uncertainty  of  re- 
sults, 60-61. 

Vertical  angles:  measurement  of  with  en- 
gineer's transit,  77-80;  with  sextant, 
91-94. 

Vertical  circles,  8. 

Year:  tropical,  38;  Julian,  39;  leap  and 
common,  39. 

Zenith,  8. 

Zenith  distance:  defined,  10;  latitude  from, 
96,  102-103;  azimuth  from,  113;  time 
from,  117-118. 

Zero  reading,  89. 


Class 


THE  ART  OF  NUMERICAL  CALCULATION 


F.  H.  SCARES 


REPRINTED  FROM  POPULAR  ASTRONOMY  No.  156. 


THE     ART    OF     NUMERICAL    CALCULATION. 


P.  H.  SEARES. 


The  circumstances  affecting  the  numerical  solution  of  prob- 
lems are  so  different  from  those  accompanying  analytical  in- 
vestigations that  the  qualifications  of  the  computer  are  quite 
distinct  from  those  of  the  mathematician.  Yet  they  include 
something  more  than  the  ability  to  combine  numbers  with 
rapidity  and  precision,  for  the  practice  of  the  art  of  computa- 
tion requires,  in  a  high  degree,  judgment,  while  the  dexterous 
manipulation  of  figures  depends  mainly  upon  a  specialized 
development  of  the  memory.  The  possession  of  one  of  these 
faculties  by  no  means  implies  the  existence  of  the  other.  Indeed 
the  extraordinary  memory  for  figures  exhibited  by  such  prodi- 
gies as  Dase,  Mondeux,  and  Inaudi  is  frequently  accompanied 
by  a  deficiency  in  those  qualities  necessary  for  th.e  practice  of 
the  art  in  its  broader  sense. 

It  is  the  purpose  of  this  paper  to  indicate  the  origin  and 
nature  of  the  requirements  of  the  computer,  and  to  formulate 
certain  details  important  for  the  numerical  solution  of  prob- 
lems occurring  in  scientific  investigations. 

I 

In  order  to  understand  the  nature  of  the  qualifications  of 
the  computer,  we  must  consider,  first,  the  origin  and  charac- 
ter of  the  data  entering  into  computations;  second,  the  dis- 
tinctive features  of  the  aids  used  by  the  calculator  for  the 
facilitation  of  his  work;  and,  finally,  the  limitations  of  mathe- 
matical expressions  when  used  as  a  basis  for  quantitative 
investigations. 

The  data  used  by  the  computer  are  usually  the  result  of  ob- 
servation and  experiment,  and  as  such,  are  affected  by  errors 
of  observation.  Errors  of  observation,  carefully  to  be  distin- 
guished from  mere  blunders,  such  as  the  incorrect  reading  of 
a  circle  by  a  whole  degree  or  an  exact  number  of  minutes, 
arise  from  a  variety  of  causes.  The  limitations  of  the  observ- 
er's senses,  defects  of  construction  and  adjustment  in  the 
instrument,  and  variations  in  the  temperature  of  the  surround- 
ing air  cause  the  final  result  to  differ  from  the  value  sought 
by  the  observer.  Certain  disturbing  factors  can  be  neutralized 
by  a  special  arrangement  of  the  observing  program  ;  others 
can  be  minimized  by  care  and  the  use  of  instruments  of  high 
precision;  but  even  when  instrument  maker  and  observer  have 
done  their  best  there  remains  in  the  final  result  an  uncertaintv 


The  Art  of  Numerical  Calculation 


beyond  control,  whose  magnitude  varies  with  the  nature  and 
circumstances  of.the  observations.  It  may  amount  to  a  con- 
siderable fraction  of  a  foot,  as  in  the  measurement  of  a  long 
line  with  the  surveyor's  chain,  or  it  may  be  only  a  few  bil- 
lionths  of  a  millimeter,  as  in  the  precise  determination  of  the 
wave  length  of  a  line  of  the  solar  spectrum.  It  may  be  a 
minute  of  arc,  as  in  the  measurement  of  the  altitude  of  a  star 
with  the  engineer's  transit,  or  only  one  or  two  tenths  of  a 
second,  as  in  the  precise  determination  of  latitude  with  the 
modern  zenith  telescope. 

The  aids  employed  by  the  computer  for  the  facilitation  of  his 
work  are  numerous.  That  most  commonly  used  is  the  ordi- 
nary logarithmic-trigonometric  table,  and  the  remarks  which 
follow  apply  particularly  to  this  contrivance,  although  the}- 
are  true,  in  a  measure,  of  the  various  other  aids  used  to  lessen 
the  labor  of  numerical  calculation. 

The  quantities  whose  numerical  values  are  contained  in  a 
table  of  logarithms  can  not  be  expressed  exactly  in  the  form 
of  i\  decimal  fraction.  The  difference  between  the  true  value 
and  the  numerical  expression  for  the  same  can  be  reduced  to 
any  assigned  limit  by  sufficiently  increasing  the  number  of 
decimals,  but,  however  great  this  number,  the  representation 
will  not  be  exact.  The  tabular  values  of  the  logarithms  are 
therefore  approximations.  The  approximations  of  a  7-place 
table  are  of  a  higher  order  than  those  of  a  5-place  table, 
but  the}'  are,  nevertheless,  approximations.  Further,  the  com- 
puter usually  requires,  not  the  printed  values  of  the  function, 
but  those  corresponding  to  intermediate  values  of  the  argu- 
ment. Thus,  he  may  desire  the  logarithm  of  the  sine  of 
26°  18'  36. "4  to  six  places  of  decimals,  a  quantity  not  given  in 
any  logarithmic  table.  The  ordinary  6-place  table  contains 
the  logarithms  of  the  trigonometric  functions  for  ever}-  10", 
and  the  required  logarithm  must  be  found  by  interpolation 
from  the  tabular  values  ior  26°  18'  30"  and  26°  18'  40". 
This  introduces  an  additional  uncertainty  for  the  interpolation 
process  is  usually  inexact.  The  total  error  in  the  interpolated 
quantity  has  a  maximum  value  of  one  unit  of  the  last  place  of 
decimals,  although  its  average  is  less.  Generalized,  this  result 
may  be  expressed  as  follows: 

E[/(A-)]  =  10-r  .  (1) 

The  symbol  E  followed  by  f  (x)  in  brackets  is  a  general  no- 
tation signifying  the  maximum  error  in  f(x).  For  the  case 


F.   H.   Scares  3 

considered,  f(x)  represents  a  quantity  interpolated  from  a  table 
of  r  places  with  the  argument  x.  For  inverse  interpolation, 
namely,  that  which  derives  the  value  of  x  corresponding  to  a 
given  f  (x),  the  maximum  error  of  the  interpolated  quantity  is 

E  [x]  =  10~T/2f'(x)  (2) 

where  f  (x)  is  the  first  derivative  of  the  tabular  function  f(x). 
From  (1)  it  will  be  observed  that  the  maximum  error  af- 
fecting the  result  of  a  direct  interpolation  is  independent  of 
the  nature  of  the  function  and  the  part  of  the  table  used,  and 
varies  only  with  the  number  of  decimals  employed.  For  in- 
verse interpolation  the  error  varies  not  only  with  the  number 
of  decimals,  but  ^also  with  the  nature  of  the  function  and  the 
magnitude  of  the  quantity  x.  As  an  illustration,  consider  the 
following  special  cases:  f(x)  =  log  x,  f(x)  —  log  sin  x,  f(x)  — 
log  cos  x,  f(x)  =•  log  tan  x.  The  application  of  (2)  gives, 

X 

when   x  is  interpolated  from  log  x,        E  [x]  —  10~r    2  .VI     ' 

p" 
log  sin  x,        E  O]  =  lu~r   -jj\r  tan  x' 

P" 
log  cos  x.        E  [x]  =  10      ~£M~         X' 

log  tan  x,        E  [x]  -  l(Tr   Ij^f  sin  2x, 

in  which  M  =  0.434  .  .  .  ,  the  modulus  of  the  common  system 
of  logarithms,  and  p"  =  206265",  the  number  of  seconds  in 
the  radian.  Thus,  the  maximum  error  in  a  number  interpo- 
lated from  its  logarithm  is  proportional  to  the  number  itself; 
and  the  maximum  errors  in  an  angle,  x,  derived  from  log  sin  x, 
log  cos  x,  and  log  tan  x  are  respectively  proportional  to  tan  x, 
cot  x,  and  sin  2x.  For  the  case  of  the  sine  and  cosine  they 
may  be  very  large  owing  to  the  presence  of  the  tangent  and 
cotangent  as  factors.  The  last  three  relations  express  analyt- 
ically the  well-known  fact  that  the  value  of  an  angle  can  be 
determined  with  greater  precision  from  the  logarithm  of  its 
tangent  than  from  that,  of  its  sine  or  cosine. 

Finally,  the  numerical  solution  of  a  problem  is  accomplished 
by  substituting  into  a  mathematical  expression  the  values  of 
the  data  corresponding  to  the  problem  in  question.  This  in- 
volves the  combination  of  a  series  of  numbers  which  are  ap- 
proximations, for,  as  we  have  seen,  both  the  data  of  observa- 
tion' and  the  quantities  derived  from  the  tables  of  logarithms 


4  The  Art  of  Numerical  Calculation 

are  uncertain.  The  final  result  must  therefore  be  an  approxim- 
ation. Its  error  arises  partly  from  the  errors  in  the  data  and 
partly  from  the  limitations  of  the  tables.  That  part  arising 
from  the  former  source  we  may  call  the  Resultant  Error  of 
Observation,  while  that  originating  in  the  limitations  of  the 
tables  is  known  as  the  Accumulated  Error  of  Calculation.  The 
magnitude  of  these  errors  depends  largely  upon  the  nature  of 
the  formula  which  expresses  the  analytical  solution  of  the 
problem.  This  may  be  such  that  the  errors  of  observation 
enter  into  the  final  result  greatly  reduced  in  magnitude.  For 
example,  in  the  determination  of  the  value  of  one  division  of 
a  scale  by  a  comparison  with  a  second  scale  of  known  length, 
we  have  for  the  required  quantity  an  expression  of  the  form 

,          rn         ro 
d  — 


in  which  rn  and  r0  are  readings  from  the  second  scale  corres- 
ponding to  the  Oth  and  the  nth  divisions  of  the  first.  What- 
ever the  magnitude  of  the  errors  of  observation,  they  enter  into 
the  final  result  by  only  I/nth  of  their  amount.  On  the  other 
hand,  quite  the  reverse  may  be  true.  Thus,  in  the  calculation 
of  the  ratio  a/b,  where  a>b,  and  h  itself  is  affected  by  an 
uncertainty,  an  error  of  observation  if  you  will,  the  result- 
ant error  of  observation  will  exceed  the  error  affecting  h 
approximately  in  the  ratio  of  a  to  62.  With  the  errors  of 
calculation  the  case  is  different.  Here  there  is  almost  al- 
ways a  certain  multiplication  of  error,  so  that  the  accumu- 
lated error  of  calculation  is  usually  in  excess  of  the  uncertain- 
ties attached  to  the  individual  numbers  which  enter  into  the 
computation;  and,  generally  speaking,  the  longer  the  calcula- 
tion, the  greater  will  be  the  accumulation  or  multiplication. 
Again,  the  solution  of  a  given  problem  is  frequently  cap- 
able of  expression  in  a  variety  of  ways.  Analytically  con- 
sidered, these  may  be  identical,  but  viewed  from  the  stand- 
point of  practical  applications,  they  may  present  the  greatest 
diversity.  For  example,  the  two  expressions 

y  —  2  sin2 1/2  x  (3) 

y  =  1  —  cos  x  (4) 

are  theoretically  equivalent,  but  when  used  for  the  calcula- 
tion of  values  of  y  corresponding  to  given  values  of  x,  they 
are  by  no  means  identical,  especially  for  values  of  x  near 
0°.  As  an  illustration,  consider  the  final  errors  resulting  from 
an  r-place  calculation  of  (3)  and  (4)  for  the  determination 


F.   H.   Scares  '  5 

of  y  corresponding  to  x  =  2°  ±  2',  where  we  may  think  of 
x  as  the  result  of  an  observation  whose  uncertainty  is  ex- 
pressed by  the  appended  quantity  ±  2'.  An  appropriate  in- 
vestigation shows  that  their  maximum  values  are 

E  [  v]  =  0.00002  +  0.003  X  10"r  ,  (3a) 

E  [j]  —  0.00002  4-  3  X  10~r .  (4a) 

The  first  terms  in  the  right  members  are  the  resultant 
errors  of  observation.  The  last  are  the  accumulated  errors 
of  calculation.  So  far  as  the  precision  to  be  obtained  with 
a  specified  number  of  decimals  is  concerned,  the  advantage  is 
obviously  in  favor  of  equation  (3). 

"With  these  facts  before  us  we  are  in  a  position  to  appre- 
ciate better  the  qualifications  required  of  the  computer.  His 
aim  must  be  so  to  arrange  the  calculation  that  the  errors 
in  the  data  and  the  errors  of  calculation  will  produce  the 
minimum  possible  effect  upon  the  final  result,  and,  at  the 
same  time,  to  derive  this  result  with  the  least  possible  ex- 
penditure of  time  and  labor.  It  is  evident  that  his  task  is 
one  of  some  complexity.  The  conditions  to  be  satisfied  are, 
to  a  certain  extent,  contradictory.  For  example,  the  accumu- 
lated error  of  calculation  can  be  reduced  to  any  desired  limit 
by  sufficiently  increasing  the  number  of  decimal  places  em- 
ployed ;  but  any  such  increase  carries  with  it  a  notable 
increase  in  the  labor  of  calculation.  On  the  other  hand,  a 
reduction  of  labor  can  often  be  brought  about  by  a  modifi- 
cation of  the  formula  to  be  calculated,  but  this  in  turn  may 
involve  a  sacrifice  of  precision. 

The  adjustment  of  these  variable  factors  to  each  other 
and  to  the  requirements  just  expressed  demands  a  nice  bal- 
ancing of  detail,  which  becomes  only  the  more  difficult  when 
it  is  considered  that  the  problems  presenting  themselves  for 
solution,  and  the  conditions  under  which  they  arise,  are  the 
most  diverse  imaginable.  It  is  obvious  that  the  computer  has 
to  deal  with  questions  whose  answers  are  not  to  be  discovered 
through  the  exercise  of  whatever  skill  he  may  possess  in  the 
manipulation  of  figures,  however  important  this  accomplish- 
ment may  be  for  the  technical  performance  of  his  labors. 
They  can  be  found  only  in  a  detailed  knowledge  of  what 
has  been  but  outlined  in  the  preceding  paragraphs.  In  addi- 
tion, the  computer  must  ever  be  upon  the  alert  \\ith  a  dis- 
criminating judgment,  if  his  work  is  to  be  consistent  in  its 


The  Art  of  Numerical  Calculation 


details,  and  economical  of    the  time  and  energy    required    for 
its  execution.  • 

II 

Leaving  now  the  consideration  of  the  subject  in  its  general 
aspects,  we  proceed  to  a  discussion  of  matters  of  more  imme- 
diate practical  significance. 

(a)          General  Arrangement  and  Procedure. 

Computations  are  most  conveniently  made  upon  cross  section 
paper  whose  squares  measure  one-fifth  or  one-sixth  of  an  inch 
on  the  side.  Before  any  figures  are  entered,  the  symbols  for 
the  quantities  to  be  combined  should  be  written  in  a  vertical 
column  at  the  left  of  the  sheet,  care  being  taken  to  bring 
together,  as  nearly  as  may  be,  those  symbols  or  arguments 
whose  numerical  values  are  to  be  combined.  Even  though 
the  same  quantity  enter  into  the  calculation  at  several  points, 
write  its  argument  but  once.  A  very  little  practice  will  make 
it  possible  to  add  or  subtract  numbers  which  are  separated 
by  several  intervening  quantities.  The  numbers  are  to  be 
written  in  a  vertical  column  immediately  to  the  right  of  the 
column  of  arguments.  If  the  same  calculation  is  to  be  per- 
formed for  a  number  of  similar  sets  of  data,  the  work  should 
appear  in  parallel  vertical  columns,  that  for  each  set  occupying 
a  column  by  itself.  In  such  a  case  do  not  complete  the  first 
column  before  beginning  the  others,  but  work  across  the  page, 
inserting  all  the  numbers  corresponding  to  any  given  argu- 
ment before  proceeding  to  the  others.  If,  however,  several 
trigonometric  functions  of  the  same  angle  are  required,  all 
should  be  interpolated  with  a  single  opening  of  the  table,  even 
though  their  symbols  occupy  widely  separated  positions  in  the 
column  of  arguments.  Further,  in  computing  for  similar  sets  of 
data,  do  not  enter  arguments  for  quantities  which  are  constant 
for  all  the  sets,  but  write  the  values  of  such  constants  on  the 
lower  edge  of  a  card  or  slip  of  paper.  This  can  be  held  above 
the  numbers  with  which  the  constants  are  to  be  united  and 
moved  along  from  column  to  column  as  the  additions  or  sub- 
tractions are  performed.  The  beginner  will  proceed  with  the 
greatest  security  by  writing  the  arguments  fully  and  complete 
ly,  although  the  experienced  computer  is  able  to  abbreviate 
the  work  by  omitting  some  of  the  arguments  and  per- 
forming the  corresponding  operations  mentally.  Thus,  it  is 
possible  to  form  the  sum  of  two  logarithms,  enter  the  table, 
and  interpolate  the  corresponding  number  without  writing 


P.  H.  Scares  7 

down  the  result  of  the  addition.  The  argument  for  the  sum 
can  therefore  be  omitted,  but  such  abbreviations  are  to  Joe 
introduced  gradually,  and  only  after  some  skill  has  been 
acquired. 

Whenever  it  becomes  necessary  to  abbreviate  a  number,  say 
to  r  places,  by  dropping  the  higher  decimals,  increase  the 
digit  of  the  rth  place  by  one  unit  when  the  neglected  quan- 
tity exceeds  one-half  a  unit  of  this  place.  If  the  decimals  neg- 
lected are  less  than  half  a  unit  of  the  ;-th  place,  they  are  to 
be  dropped  without  change  in  that  place.  When  the  neglected 
part  is  exactly  a  half  unit  of  the  rth  place,  it  is  a  good  rule 
to  increase  the  digit  of  the  rth  place  by  one  unit,  in  case  that 
digit  is  odd ;  otherwise,  drop  the  higher  places  without  change 
in  the  rth  place.  Errors  arising  from  the  abbreviation  will 
thus  tend  to  neutralize  each  other  in  the  long  run. 

(b)  Aids    to    the    Computer. 

Machines  for  addition,  multiplication  etc.  Their  operation 
is  so  simple  that  they  require  no  special  treatment  in  this 
place.  Their  construction  is  such  that  there  is  no  accumulated 
error  of  calculation,  unless  the  quantities  involved  are  abbre- 
viated by  dropping  higher  decimal  places. 

The  sliderule.  In  effect,  this  instrument  is  a  graphical  table 
of  logarithms.  In  its  usual  form,  the  accumulated  error  of 
calculation  generally  amounts  to  a  few  units  of  the  fourth 
place  of  decimals.  It  is,  therefore,  in  nowise  a  substitute  for 
tables  of  logarithms  of  5,  6,  and  7,  or  even  4  places.  It  is 
extremely  convenient  for  certain  classes  of  computation,  but 
many  experienced  computers  maintain  that  properly  con- 
structed tables  of  logarithms  give  more  satisfactory  results. 
In  any  case,  its  continued  use  results  in  a  strain  upon  the 
eyes  far  greater  than  that  accompanying  the  use  of  well 
printed  tables. 

Multiplication  tables.  The  best  are  the  Rechentafeln  oi 
Crelle,  published  by  Reimer  of  Berlin.  These  tables  give 
directly  the  exact  products  of  numbers  of  three  figures  or 
less;  and  can  be  used  for  the  determination  of  products  of 
numbers  of  any  magnitude.  They  can  also  be  used  for  divi- 
sion, most  conveniently,  when  the  divisor  is  of  three  figures 
or  less.  Their  only  objection  is  their  bulk.  They  should  be 
in  the  hands  of  every  computer. 

Logarithmic-trigonometric  tables.  Those  most  generally 
useful  are  of  five  places  of  decimals,  although'  3,  4,  6,  and 


8  The  Art  of  Numerical  Calculation 

7-place  tables. are  also  frequently  required,  and  should  be 
within  reach  of  all  who  have  to  deal  with  astronomical  or 
geodetic  calculations.  In  purchasing  tables,  care  should  be 
exercised,  for  many  are  badly  arranged  and  unfit  for  the 
purpose  for  which  they  are  intended.  With  the  exception  of 
3-place  tables,  those  not  giving  the  differences  of  the  adjacent 
logarithms,  at  least  for  the  tables  of  trigonometric  functions, 
should  be  avoided.  The  same  is  true  of  those  not  containing 
auxiliary  tables  of  proportional  parts.  The  tabulation  of 
the  logarithms  of  the  trigonometric  functions  to  six  places 
of  decimals  for  every  minute  of  arc,  only,  is  likewise  a  bad 
arrangement.  It  is  also  important  that  the  tables  for  the 
sine  and  cosine  should  not  be  separated  from  those  of  the 
tangent  and  cotangent.  And,  finally,  it  is  desirable  to  select 
tables  containing  addition-subtraction  logarithms.  There  are 
numerous  other  points  of  minor  importance,  but  a  more 
detailed  discussion  can  be  replaced  by  the  following  list  of 
satisfactory  tables.  The  list  does  not  pretend  to  be  complete. 
Four-place  tables : 

Slichter,   Macmillan  ; 

Bremiker,   Weidmannsche  Buchhandlung,  Berlin. 
Five-place    tables : 

Becker,  Tauchnitz,  Leipzig; 
Gauss,   Strien,   Halle; 
Albrecht,   Stankiewicz,   Berlin; 
Newcomb,  H.   Holt  &  Co.,  New  York  ; 
Hussey,  Allyn  &  Bacon,  Boston  ; 
Bremiker,  Weidmannsche  Buchhandlung,  Berlin. 
Six-place  tables: 

Bremiker,   edited   by    Albrecht,   Nicolaische 

Verlags-Buchhandlung,   Berlin. 
Seven-place  tables : 

Vega,  edited  by  Bremiker,  Weidmannsche  Buchhandlung. 

This  is   the  best    7-place  table. 
Bruhns,   Tauchnitz,    Leipzig. 

In  Bremiker's  4  and  5-place  tables,  the  arguments  for  the 
trigonometric  functions  are  expressed  in  decimals  of  a  degree, 
the  intervals  being  0°.l  and  0°.01,  respectively,  for  the  body 
of  the  tables. 

It  is  assumed  that  the  student  is  familiar  with  the  funda- 
mental principles  underlying  the  construction  and  use  of  the 
ordinary  logarithmic-trigonometric  tables.  The  details  of  their 
usage  can  therefore  be  dismissed  with  the  following  precepts: 


F.  H.  Scares  9 

(1)  Do  not  use  negative  characteristics.     When  such  occur, 
increase  them  by  ten,  and  operate  as  though  a  minus  ten  were 
written    after  the  logarithm.     When  two  such  logarithms  are 
added,  the  sum    will  have  an  appended   minus  twenty,  which 
should    be    reduced    to  minus  ten,  dropping  at  the  same  time 
ten  units  from  the  characteristic. 

(2)  In  case  the  number  corresponding  to  a  given  logarithm 
is  negative    indicate  that  fact  by  writing  a  subscript    n  after 
the  logarithm.    In  combining  a  number  of  logarithms,  affix  a 
subscript  n  to  the  result  when  the  number  of  w-logarithms  is 
odd.  If  this  is  even,  the  resulting  logarithm  needs  no  subscript. 

(3)  Derive    the    logarithm  of  the  secant  and  cosecant  from 
those   ot    the  cosine  and  sine,  respectively,  by  subtracting  the 
latter  from    zero.     This    is    most   easily  accomplished  by  sub- 
tracting   each  digit  of  the  logarithm  from  9,  proceeding   from 
left  to  right,  until  the  last  is  reached,  which  is  to  be  subtracted 
from  10. 

(4)  Interpolate  all  the  functions  required  for  any  given  angle 
with  a  single  opening  of  the  table. 

(5)  In  the  formation  of  powers    of  numbers,  care  must  be 
exercised   when    the    power    is  fractional,   and  the  number  less 
than  unity.     After  the  logarithm  of  the  number  has  been  mul- 
tiplied by  the  power,    p,  the  appended  characteristic,  which  is 
normally  —10,  will  be  —  lOp.   This  must  be  reduced  to  —10  by 
adding  10(1  —  p)  to  the  characteristic  proper  and  subtracting 
the    same  quantity    from  the  characteristic    appended    to    the 
result  of  the  multiplication. 

Addition-subtraction  logarithms.  The  purpose  of  these  tables 
is  to  determine  the  logarithm  of  a  +  b  when  the  logarithms  of 
a  and  h  are  given.  The  following  illustrates  the  principle 
underlying  their  construction  and  use: 

Let 

4  =  log  N  , 

B  ='log  (AT  +  1), 

where    N  represents    any    number.      The    addition-subtraction 
logarithmic  table  contains  the  values  of  B  tabulated  with  the 
argument  A. 
Now  suppose 


whence 

A  =  log  b  —  log  a  , 

and 


10  The  Art  of  Numerical  Calculation 

B  -  log  (-4   1)  =  log  (a  +  b)  -  log  a, 

• 

or 

log  (a  +  b)  —  log  a  +  B. 

This  is  the  fundamental  equation  for  addition.    The  procedure 
is  as  follows: 
Form  A  =  log  b  —  log  a. 

Interpolate  B  with  A  as  argument. 
Then, 

log  (a  +  b)  =  loga  +  B. 

Again,   let 


b 
whence 

A  =  log  (a  —  fo)  —  log  b, 

and 

a 
B  =  log  -7-  —  log  a  —  log  />. 

These  are  the  fundamental  equations    for  subtraction.      The 
procedure  is  as  follows : 
Form 

B  =  log  a  —  log  b. 

Interpolate  A  with  B  as  argument. 
Then, 

log  (a  —  b)  =  log  b  -j-  A. 

The  arrangement  of  the  tables  assumed  a>h  for  the  solution 
of  both  the  addition  and  the  subtraction  problem.  The  appli- 
cation of  these  tables  involves  the  performance  of  one  subtrac- 
tion, one  addition,  and  one  interpolation.  The  use  of  the 
ordinary  logarithmic  table  for  the  derivation  of  the  same 
result  involves  three  interpolations  and  one  addition  or  sub- 
traction. Further,  as  an  illustration,  in  the  5-place  tables  of 
Gauss,  the  ordinary  table  covers  18  pages  while  the  addition- 
subtraction  table  covers  but  12,  of  which  only  4V&  are  neces- 
sary for  the  addition  problem.  Finally,  it  can  be  shown  that 
the  uncertainty  of  a  result  derived  from  the  addition-subtrac- 
tion table  is  less  than  that  accompanying  the  use  of  the 
ordinary  table.  From  every  standpoint,  therefore,  whether 
that  of  the  number  of  operations  to  be  performed,  the  number 
of  pages  to  be  thumbed,  or  the  accuracy  of  the  final  result, 
the  advantage  is  in  favor  of  the  addition-subtraction  table. 

Tables  of  squares,  cubes,  etc.  Special  tables.   Barlow's  Tables, 


F.  H.  Scares  H 

containing  the  squares,  cubes,  square  roots,  cube  roots,  and 
reciprocals  of  all  integers  up  to  10,000,  is  one  of  the  most 
convenient.  The  roots  are  given  to  seven  places  of  decimals, 
and  the  reciprocals  partly  to  nine  and  partly  to  ten  places. 
Some  of  the  5-place  logarithmic  tables,  such  as  those  ot  Gauss 
and  Albrecht,  also  contains  tables  of  squares.  Any  table  of 
squares  or  cubes  can  be  used  inversely  for  the  derivation  of 
square  and  cube  roots. 

In  addition  to  the  various  aids  mentioned  there  are  innumer- 
able special  tables  designed  for  the  solution  of  special  problems. 
Almost  any  problem  which  has  to  be  solved  repeatedly  for 
different  sets  of  data  can  be  simplified  through  the  use  of 
specially  constructed  tables.  In  this  connection  there  is  abun- 
dant opportunity  for  the  exercise  of  ingenuity  and  skill  on  the 
part  of  the  computer. 

(c)  Resultant  Error  of  Observation,  Accumulated  Error  of 
Calculation.  Number  of  Decimal  Places. 

The  estimation  of  the  effect  on  the  final  result  due  to  un- 
certainties in  the  data,  in  other  words,  the  evaluation  of  the 
resultant  error  of  observation,  is  most  conveniently  made  by 
means  of  the  relation  obtained  by  differentiating  the  formulae 
to  be  solved  with  respect  to  the  final  result  and  the  quantities 
whose  values  are  given.  The  substitution  of  the  uncertainties 
in  the  data  for  the  corresponding  differentials  in  this  express- 
ion leads  to  a  knowledge  of  the  numerical  value  of  the  differ- 
ential of  the  final  result,  which  may  be  taken  as  the  resultant 
error  of  observation. 

The  determination  of  the  maximum  possible  value  of  the 
accumulated  error  of  calculation  for  any  given  set  of  formulas 
is  a  more  or  less  complicated  process.  Since,  however,  the 
accumulated  error  seldom,  if  ever,  reaches  its  maximum,  a 
knowledge  of  its  average  magnitude  is  of  more  practical  im- 
portance. This  varies  with  the  character  of  the  equations  to 
be  solved,  and  its  exact  evaluation  presents  some  difficulty. 
But  for  formulas  containing  no  critical  features,  such  as  ab- 
normally large  multipliers  or  small  divisors,  or  differences 
defined  by  two  relatively  large  and  nearly  equal  quantities,  or 
angles  to  be  interpolated  from  sines  or  cosines,  the  following, 
based  upon  the  theory  of  probabilities,  gives  an  approximate 
expression  of  the  average  uncertainty  in  the  logarithm  of  a 
result : 

tfi.  =0.4X  lO'Vn 


12 


The  Art  of  Numerical  Calculation 


where  r  is  the  .number  of  decimal  places  employed,  and  n  the 
number  of  quantities  involved  in  the  calculation.  If  the  final 
result  is  a  number,  AT,  its  approximate  average  uncertainty 
will  be  given  by 


or,  if  an  angle,  by 


Uj,  =  0.4  X  10~r  206265"  \/ n. 


(6) 


(7) 


The  following  table  shows  the  results  given  by  (5),  (6),  and 
(7)  for  tables  of  3  to  7  places,  n  being  equal  to  unity.  To 
obtain  the  accumulated  error  of  calculation  for  any  given  case, 
it  is  only  necessary  to  multiply  the  proper  tabular  value  by 
the  square  root  of  the  number  of  quantities  entering  into  the 
calculation. 


1     No.  of 
Decimals  =  r 

n=  1 

*7L                   ]                       Ux                \               U± 

3 
4 
5 
6 

7 

0.4  X  10~34 
0.4  X  10 
0.4  X  1(T° 
0.4  X  10 
.  0.4  X  10~' 

10~3  N 
1C"  N 
10-°AT 

10-"  jv 
10-'  .v 

1/4 

8" 
0.8 
0.08 
0.008 

Experience  shows  that  these  results  are  in  close  agreement 
with  the  average  values  actually  occurring  in  practice. 

The  determination  of  the  number  of  decimal  places  to  be  used 
in  any  given  calculation  is  a  matter  of  great  importance.  If 
the  number  chosen  is  too  small,  the  precision  of  the  data  will 
be  sacrificed.  If  too  large,  much  unnecessary  labor  will  be 
expended— just  how  much,  is  suggested  by  the  fact  that  the 
relative  amounts  of  time  required  to  execute  a  calculation 
with  4.  5,  6,  and  7  places  of  decimals  are  approximately  ex- 
pressed by  the  numbers  1,  2,  3,  and  5,  respectively,  i.  e.,  five 
times  as  much  labor  is  required  to  complete  a  given  calcula- 
tion with  7-place  logarithms  as  would  be  required  if  only 
4-place  tables  were  used.  In  practice,  the  number  actually 
to  be  employed  is  usually  determined  by  the  accuracy  of  the 
given  data.  If  this  is  to  be  used  to  its  full  advantage,  a 
sufficient  number  of  decimals  must  be  employed  to  make  the 
accumulated  error  of  calculation  small  as  compared  with 
the  resultant  error  of  observation.  Having  determined  the 
amount  of  the  latter,  the  above  table  affords  such  indica- 


necessarily  small  as  compared  with  the  resultLt  er  or  of 
ob ervation  showing  that  the  use  of  6-place  tables  woud  in- 
volve a  needless  amount  of  labor.  The  second  value,  on  the 
other  hand  indicates  that  5-place  tables  will  entail  he  m Li- 
mum  of  labor  consistent  with  the  precision  desired 

gain,  required  the  number  of  decimals  necessary  for  a 
calculation  involving  25  logarithms,  in  which  the  resultant 
error  of  observation  is  0.0001,  the  result  itself  being  a  num- 
3er  whose  approximate  value  is  100.  The  expression  for  the 
accumulated  error  ot  calculation  is 

Uy[  —  10  r  ><  100  X  5  =  5  X  102~r- 

To  make  this  small  as  compared  with  the  resultant  error 
of  observation,  r  must  be  taken  equal  to  7. 

For  formulae  free  from  the  critical  features  mentioned  in 
the  second  paragraph  of  this  section,  the  resultant  error  of 
observation  will  usually  be  of  the  order  of  the  uncertainties 
in  the  data.  The  choice  of  the  number  of  decimals  is  then 
very  simple.  If  the  data,  consists  of  numbers,  it  is  only  nec- 
essary to  choose  a  number  of  decimals  greater  by  one  than 
the  number  of  significant  figures  in  the  given  quantities. 
If,  on  the  other  hand,  the  data  consists  of  angles,  a  glance 
at  the  tabular  values  of  £7A  affords  the  necessary  information. 

The  above  suggestions  by  no  means  cover  all  the  cases 
which  may  arise  in  practice,  but  they  give  an  indication  as 
to  the  general  method  of  procedure. 

(d)  The    Adaptation    of  Formulae. 

The  following  general  suggestions  indicate  the  more  im- 
nortant  points  to  be  borne  in  mind. 

m  Whenever  possible,  transform  equations  containing 
^fFWpnces  of  terms  into  expressions  containing  only 
Quotients.  Thus,  the  equations 


14  The  Art  of  Numerical  Calculation 

sin  5  =  cos  z  sin  <£  —  sin  z  cos  </>  cos  .A, 
cos  8  cos  t  •=  cos  z  cos  </>  +  sin  z  sin  0  cos  A,  (8) 

cos  5  sin  t  =  sin  z  sin  /I, 

defining  8  and  t  in  terms  of  z,  <j>,  and  A  can  be  reduced  to 
the  form 

sin  5  —  777  sin  ( </>  —  M )  , 

cos  5  cos  t  =  m  cos(</>  —  M),  (9) 

cos  5  sin  t  =  sin  z  sin  A, 

by  introducing  the  auxiliaries  m  and   Af   defined  b}r 

m  sin  M  =  sin  z  cos  A  , 

(10) 

777  COS  M  =  COS  Z  . 

The  solution  of  (10)  and  (9)  thus  replaces  the  solution  of  (8). 
Although  the  number  of  equations  involved  in  (9)  and  (10) 
is  five  as  against  three  in  (8),  the  calculation  is  usually  simpler 
in  arrangement  and  control. 

(2)  Whenever  possible,  calculated    angles  should    be  deter- 
mined from  the  tangent.   This  does  not  mean  that  the  formulae 
are  necessarily    to  be  arranged  so  as  to  express  explicitly  the 
tangent    of  a  required    angle.      As  an    illustration,   M  in  (10) 
will    be  determined    from    its  tangent,  derived  by  subtracting 
log  723  cos  M  from  log  m  sin  M,  although  the  equations  do  not 
express  the  tangent  of  M  explicitly.      The  same  is  true  of  the 
determination  of  t  from  the  last  two  of  (9).     It  is  possible  to 
replace  equations  (9)  and  (10)  by  a  single  group  of  three  equa- 
tions giving  directly   the   tangents  of  M,  t,   and   8;   but  this  is 
not  to  be  recommended,  for    there  is  no  saving  in  labor,   and 
the  use  of  (9)  and  (10)   as    they    stand    introduces    symmetry 
into  the  arrangement  of  the  work,  and  simplifies  the  determin- 
ation of  the  quadrants  and  the  control  of  the  calculation. 

(3)  Avoid    formulas  expressing  a  quantity  as  the    difference 
of  two  relatively  large  and  nearly  equal  numbers.      The  com- 
parison   of  equations    (3)   and  (4)  made  in  a  previous  section 
illustrates    the    disadvantage    connected    with    expressions    of 
this    type. 

(4)  When    it  is    necessary  to  determine  a  quantity  differing 
but    little    from    a    second    quantity    whose    value    is    known, 
arrange  the  formulae  in  such  a  way  as  to  express  the  difference 
of  the    two.      The    calculated   difference  applied  to  the  known 
quantity  then  gives  the  desired  result.      Developments  in  series 
are  frequently  useful  in  this  connection.      Thus,  the    geocentric 


F.  H.  Scares  15 

latitude,   <£',   is  given  by  the  relation 

tan  <f>'  =  (  I  —  e2)  tan  <f>,  (11) 

in  which  <£'  is  the  astronomical  latitude,  and  e,  the  eccentricity 
of  a  meridional  section  of  the  Earth.  The  numerical  value  of 
the  latter  is  approximately  0.08,  whence  it  follows  that  <£' 
differs  but  little  from  <£.  In  accordance  with  the  above  men- 
•tioned  principle,  (11)  is  replaced  for  the  purpose  of  numerical 
calculation  by  the  following"  equivalent  expression 

<t>'  —  <t>  --  —  690".65  sin  2<f>  -f-  l".16  sin  4<£  .  .  .  (12) 

in  which  the  neglected  terms  are  insensible.  The  first  term  in 
the  right  member  of  (12)  calculated  with  5-place  and  the  sec- 
ond, with  3-plac.e  logarithms,  gives  the  same  precision  as  7- 
place  tables  used  in  connection  with  (11). 

(5)  Calculations  can  frequently  be  much  simplified  by  the 
use  of  approximate  formulae.  It  is  obviously  permissible  to 
neglect  those  terms  in  an  equation  whose  numerical  values 
are  small  as  compared  with  the  resultant  error  of  observation. 
One  of  the  most  common  methods  of  introducing  such  simpli- 
fications. consists  in  the  substitution  of  the  first  terms  of  the 
developments  in  series  of  the  sine,  cosine,  and  the  tangent  of 
small  angles  for  the  trigonometric  functions  themselves.  Since 

*3  -i 

sin  x  =  x  —   6    +  ... 

'          .-  (13) 


tan  x  —  x  +  *'    -f-  ... 

O 

it  follows  that  the  errors  resulting  from  the  substitution  men- 
tioned will  be  of  the  order  ot  xs/6,  x2/>,  and  xah,  respectively. 
The  evaluation  of  the  errors  in  any  given  case  is  readily 
accomplished  by  means  of  the  approximate  relations 

A2  —  (-Y  in  degrees)2  X  1'  i 

(14) 

A-3  =  (x  in  degrees)3  X  1"  • 


Thus,    for    x  =  15',  the  substitution    of  x  for  sin  *  introduces 
the  error 

3  i" 

384  ' 


i  /i  \ 

_  [    —    I 

6\-T./ 


16  The  Art  of  Numerical  Calculation 

For  the  same  value  of  x,  the  substitution  of  1  for  cos  x  pro- 
duces the  error 

1  /  1  V  1' 

2Y4  )  xl/==  "32  =2"  '          approximately  . 

It  will  be  noted  that  the  use  of  the  suggestion  in  (4)  will 
frequently  make  possible  substitutions  of  this  character,  for  an 
arrangement  of  the  formulae  expressing  the  difference  between 
the  required  quantity  and  a  nearly  equal  known  quantity  often- 
times introduces  the  trigonometric  functions  of  small  angles. 

As   an  illustration,  the    equation 

sin  h  =  cos  TT  sin  <f>  +  sin  TT  cos  <f>  cos  t  (15) 

expresses  the  relation  between  the  north  polar  distance  of  a 
star,  TT,  its  hour  angle,  t,  its  altitude  h,  and  the  latitude  of 
the  place,  <£.  The  latitude  can  be  calculated  .when  the  remain- 
ing quantities  are  known.  One  of  the  most  convenient  methods 
results  from  an  application  of  (15)  to  the  star  Polaris.  Since 
the  latitude  equals  the  altitude  of  the  celestial  pole  above  the 
horizon,  and  since  this  latter  differs  at  most  by  1°  12'  from 
the  altitude  of  Polaris,  it  is  desirable  to  express  (15)  as  a 
function  of  H  =  <£  —  h.  The  resulting  equation  in  H  is 

sin  H  •=.  —  sin  IT  cos  t  -j-  tan  <£  (cos  H  —  cos  IT  ) .  (16) 

Since  H  ^  TT  =  1°  12',  (16)  can  be  replaced  by  the  approxim- 
ate relation 

7f2 

H  =  —  TT  cos  t  -\-  -^  tan  <j>  sin2  t ,  (17) 

in  which  terms  of  the  order  of  ?r3  or  higher  have  been  neglected. 
The  maximum  error  of  (17)  is  V  or  2".  The  appearance  of 
<£  in  the  last  term  of  (17)  seems  to  require  a  knowledge  of 
the  quantity  for  whose  determination  the  equation  has  been 
designed,  viz.,  the  latitude.  But,  since  the  coefficient  of  this 
term  is  very  small,  a  rough  approximation  for  <f>,  which  can 
be  obtained  in  a  variety  of  ways,  is  all  that  is  required. 

(e)        Control  and  Checking  of  Computations. 

The  acquirement  of  accuracy  in  the  performance  of  numerical 
calculations  is  largely  a  matter  of  intelligent  practice.  The 
mistakes  of  the  beginner  are  due  in  part  to  an  inability  to 
concentrate  his  attention  upon  the  large  number  of  details 
involved  in  even  a  relatively  short  calculation.  With  experi- 
ence, the  more  frequently  occurring  operations  are  reduced  to 
a  more  or  less  mechanical  process.  This  reduces  the  strain 


F.  H.  Scares  17 

upon  the  attention,  and  thereby  lessens  the  liability  to  error. 
But  this  liability  is  never  entirely  removed,  and  the  most  skill- 
ful computer  occasionally  makes  mistakes.  '  It  is  therefore 
essential  that  all  calculations  should  be  controlled.  The 
importance  of  this  cannot  be  too  strongly  emphasized,  especi- 
ally for  the  beginner.  The  methods  of  checking  are  numerous. 
Generally  each  type  of  problem  requires  its  own  special  method 
of  treatment.  One  of  the  most  satisfactory  methods  of  control 
is  afforded  by  the  derivation  of  a  result  from  two  independent 
sets  of  formulas.  There  are  many  cases,  however,  in  which 
the  application  of  such  a  test  is  impossible.  Another  is  the 
so-called  method  of  differences,  which  can  be  applied  when  the 
same  set  of  formulae  is  to  be  calculated  for  several  uniformly 
varying  sets  of  data.  In  such  cases  the  final  results,  and  in- 
deed, the  numerical  values  of  any  given  quantity  in  the  calcu- 
lation, present  a  systematic  variation  as  one  passes  successively 
from  one  set  of  data  to  another.  The  test  is  applied  b}r  form- 
ing the  successive  differences  of  the  numerical  values  of  the 
final  results  or  of  such  quantities  as  it  is  deemed  necessary  to 
control.  A  numerical  error  in  one  or  more  of  the  separate 
calculations  reveals  itself  through  an  irregularity  in  the  varia- 
tion of  the  differences.  The  control  is  a  searching  one  and  is 
capable  of  bringing  to  light  even  the  neglected  decimals  form- 
ing a  part  of  the  accumulated  error  of  calculation.  The 
arithmetic  mean  of  a  series  of  numbers  can  be  checked  by 
forming  the  differences  between  the  mean  and  the  individual 
numbers,  regard  being  paid  to  the  algebraic  sign.  If  the  calcu- 
lated mean  is  correct,  the  sum  of  the  positive  differences  will 
equal  that  of  the  negative  differences. 

Other  methods  of  checking  will  readily  suggest  themselves 
to  the  computer  after  a  moderate  amount  of  experience,  but 
whatever  the  method,  he  must  constantly  be  on  his  guard  not 
to  place  too  much  confidence  in  a  mere  agreement  of  numerical 
results,  for  no  process  of  checking  is  absolute.  An  agreement 
in  numerical  results  signifies  at  most  a  certain  probability 
that  a  calculation  has  been  correctly  performed.  This  proba- 
bility may  be  large  or  small  according  to  the  nature  of  the 
test,  but  it  never  becomes  equal  to  certainty.  For  example, 
the  method  of  differences  affords  an  invaluable  control  in  so 
far  as  isolated  errors  are  concerned,  but  it  is  quite  incapable 
of  discovering  systematic  errors  affecting  similarly  all  of  the 
separate  calculations.  It  is  therefore  desirable  that  several 
different  tests  should  be  applied.  For  those  rare  cases  in 


18  The  Art  of  Numerical  Calculation 

which  all  ordinary  processes  of  checking  become  impossible, 
the  calculation  should,  be  repeated  by  a  second  computer, 
working  quite  independently  of  the  first ;  and  failing  this,  by 
the  original  computer  himself.  But  this  last  method  should 
be  adopted  only  as  a  final  resort,  and  then  only  after  a  con- 
siderable interval  of  time  has  elapsed,  for  it  is  a  well-known 
fact  that  an  error  once  committed  is  very  likely  to  recur  in  the 
repeated  calculation. 

The  preceding  remarks  relate  particularly  to  the  detection 
of  errors  already  committed.  Something  should  also  be  said  as 
to  the  methods  of  preventing  errors.  The  most  important 
things  are  care,  attention,  and  deliberation.  The  systematic 
arrangement  of  the  work  explained  under  (a)  contributes  much. 
The  method  of  combining  numbers  is  of  importance.  When 
tw°  numbers  are  to  be  added  or  subtracted,  the  combination 
should  be  made  from  left  to  right.  With  a  little  practice  this 
method  affords  an  increase  in  both  accuracy  and  rapidity  as 
compared  with  the  usual  process.  With  the  beginner  each 
result  should  be  verified  as  the  calculation  progresses,  always 
by  an  independent  method,  if  possible.  Thus,  additions,  in  the 
case  of  two  numbers,  can  be  performed  both  from  left  to 
right  and  from  right  to  left;  for  more  than  two  numbers,  by 
adding  from  the  top  downward  and  from  the  bottom  up; 
subtractions  can  be  controlled  by  adding  the  difference  to  the 
subtrahend ;  interpolations  of  trigonometric  functions,  by  a 
comparison  of  the  difference  of  the  logarithms  of  the  sine  and 
cosine  with  the  logarithm  of  the  tangent.  Finally,  the  com- 
puter should  be  on  the  watch  for  such  impossible  results  as 
values  of  a  sine  or  cosine  greater  than  unity,  and  the  occurr- 
ence of  negative  values  for  essentially  positive  numbers. 

Under  no  circumstances  should  these  details  be  neglected 
unless  the  computer  has  acquired  a  very  considerable  skill,  for 
it  is  a  matter  of  experience  that  the  location  and  correction 
of  errors  in  a  completed  calculation  consumes  far  more  time 
than  is  required  for  the  execution  of  the  work  with  that  de- 
liberation which  assures  some  degree  of  accuracy  in  the  final 
result.  In  addition,  the  consciousness  of  having  exercised  all 
possible  care  gives  a  feeling  of  security  which,  in  the  attempt 
to  secure  freedom  from  errors,  is  a  psychological  factor  of  no 
small  importance. 

Laws  Observatory, 
Columbia,  Mo. 


